diff --git "a/1001.0030.txt" "b/1001.0030.txt" new file mode 100644--- /dev/null +++ "b/1001.0030.txt" @@ -0,0 +1,9745 @@ +arXiv:1001.0030v2 [math.CO] 28 Feb 2012CYCLIC SIEVING FOR GENERALISED NON-CROSSING +PARTITIONS ASSOCIATED WITH COMPLEX REFLECTION +GROUPS OF EXCEPTIONAL TYPE — THE DETAILS +Christian Krattenthaler†andThomas W. M ¨uller‡ +†Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien, +Nordbergstraße 15, A-1090 Vienna, Austria. +WWW:http://www.mat.univie.ac.at/ ~kratt +‡School of Mathematical Sciences, +Queen Mary & Westfield College, University of London, +Mile End Road, London E1 4NS, United Kingdom. +WWW:http://www.maths.qmw.ac.uk/ ~twm/ +Dedicated to the memory of Herb Wilf +Abstract. We prove that the generalised non-crossing partitions associated with +well-generated complex reflection groups of exceptional type obe y two different cyclic +sieving phenomena, as conjectured by Armstrong, and by Bessis a nd Reiner. This +manuscript accompanies the paper “Cyclic sieving for generalised non-crossing parti- +tions associated with complex reflection groups of exceptio nal type” [arχiv:1001.0028 ], +for which it provides the computational details. +1.Introduction +In his memoir [2], Armstrong introduced generalised non-crossing partitions asso- +ciated with finite (real) reflection groups, thereby embedding Krew eras’ non-crossing +partitions [22], Edelman’s m-divisible non-crossing partitions [12], thenon-crossing par- +titions associated with reflection groups due to Bessis [6] and Brady and Watt [10] into +one uniform framework. Bessis and Reiner [9] observed that Arms trong’s definition can +be straightforwardly extended to well-generated complex reflection groups (see Section 2 +for the precise definition). These generalised non-crossing partit ions possess a wealth +of beautiful properties, and they display deep and surprising relat ions to other combi- +natorial objects defined for reflection groups (such as the gene ralised cluster complex +2000Mathematics Subject Classification. Primary 05E15; Secondary 05A10 05A15 05A18 06A07 +20F55. +Key words and phrases. complex reflection groups, unitary reflection groups, m-divisible non- +crossing partitions, generalised non-crossing partitions, Fuß–Ca talan numbers, cyclic sieving. +†Research partially supported by the Austrian Science Foundation F WF, grants Z130-N13 and +S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics +and Probabilistic Number Theory.” +‡Research supported by the Austrian Science Foundation FWF, Lise Meitner grant M1201-N13. +12 C. KRATTENTHALER AND T. W. M ¨ULLER +of Fomin and Reading [13], or the extended Shi arrangement and the geometric multi- +chains of filters of Athanasiadis [4, 5]); see Armstrong’s memoir [2] and the references +given therein. +Ontheotherhand, cyclic sieving isaphenomenonbroughttolightbyReiner, Stanton +and White [30]. It extends the so-called “( −1)-phenomenon” of Stembridge [34, 35]. +Cyclic sieving can be defined in three equivalent ways (cf. [30, Prop. 2.1]). The one +which gives the name can be described as follows: given a set Sof combinatorial +objects, an action on Sof a cyclic group G=/an}bracketle{tg/an}bracketri}htwith generator gof ordern, and +a polynomial P(q) inqwith non-negative integer coefficients, we say that the triple +(S,P,G)exhibits the cyclic sieving phenomenon , if the number of elements of Sfixed +bygkequalsP(e2πik/n). In [30] it is shown that this phenomenon occurs in surprisingly +many contexts, and several further instances have been discov ered since then. +In [2, Conj. 5.4.7] (also appearing in [9, Conj. 6.4]) and [9, Conj. 6.5], Ar mstrong, +respectively Bessis and Reiner, conjecture that generalised non- crossing partitions for +irreducible well-generated complex reflection groups exhibit two diffe rent cyclic sieving +phenomena (see Sections 3 and 7 for the precise statements). +According to the classification of these groups due to Shephard an d Todd [32], there +are two infinite families of irreducible well-generated complex reflectio n groups, namely +the groups G(d,1,n) andG(e,e,n), wheren,d,eare positive integers, and there are 26 +exceptional groups. For the infinite families of types G(d,1,n) andG(e,e,n), the two +cyclic sieving conjectures follow from the results in [19]. +The purpose of the present article is to prove the cyclic sieving conj ectures of Arm- +strong, and of Bessis and Reiner, for the 26 exceptional types, t hus completing the +proof of these conjectures. Since the generalised non-crossing partitions feature a pa- +rameterm, from the outset this is nota finite problem. Consequently, we first need +several auxiliary results to reduce the conjectures for each of t he 26 exceptional types +to afiniteproblem. Subsequently, we use Stembridge’s Maplepackagecoxeter [36] +and theGAPpackageCHEVIE[14, 28] to carry out the remaining finitecomputations. +It is interesting to observe that, for the verification of the type E8case, it is essential +to use the decomposition numbers in the sense of [17, 18, 20] becau se, otherwise, the +necessary computations would not be feasible in reasonable time with the currently +available computer facilities. We point out that, for the special case where the afore- +mentioned parameter mis equal to 1, the first cyclic sieving conjecture has been proved +in a uniform fashion by Bessis and Reiner in [9]. (See [3] for a — non-unifo rm — proof +of cyclic sieving for non-crossing partitions associated with realreflection groups under +the action of the so-called Kreweras map, a special case of the sec ond cyclic sieving +phenomenon discussed in the present paper.) The crucial result on which the proof of +Bessis andReiner is based is(5.5) below, andit plays animportant role in our reduction +of the conjectures for the 26 exceptional groups to a finite prob lem. +Our paper is organised as follows. In the next section, we recall the definition of +generalised non-crossing partitions for well-generated complex re flection groups and of +decomposition numbers in the sense of [17, 18, 20], and we review so me basic facts. +The first cyclic sieving conjecture is subsequently stated in Section 3. In Section 4, we +outline an elementary proof that the q-Fuß–Catalan number, which is the polynomial +Pin the cyclic sieving phenomena concerning the generalised non-cros sing partitions +for well-generated complex reflection groups, is always a polynomial with non-negative +integer coefficients, asrequired bythedefinitionofcyclic sieving. (T he readerisreferredCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 3 +to the first paragraph of Section 4 for comments on other approa ches for establishing +polynomiality with non-negative coefficients.) Section 5 contains the a nnounced auxil- +iary results which, for the 26 exceptional types, allow a reduction o f the conjecture to a +finite problem. The remaining case-by-case verification of the conj ecture is then carried +out in Section 6. The second cyclic sieving conjecture is stated in Sec tion 7. Section 8 +contains the auxiliary results which, for the 26 exceptional types, allow a reduction of +the conjecture to a finite problem, while Section 9 contains the rema ining case-by-case +verification of the conjecture. +2.Preliminaries +Acomplex reflection group isa groupgeneratedby(complex) reflections in Cn. (Here, +a reflection is a non-trivial element of GLn(C) which fixes a hyperplane pointwise and +which hasfiniteorder.) Wereferto[24]foranin-depthexpositionof thetheorycomplex +reflection groups. +Shephard and Todd provided a complete classification of all finitecomplex reflection +groups in [32] (see also [24, Ch. 8]). According to this classification, a n arbitrary +complex reflection group Wdecomposes into a direct product of irreducible complex +reflection groups, acting on mutually orthogonal subspaces of th e complex vector space +onwhichWisacting. Moreover, thelistofirreduciblecomplexreflectiongroups consists +of the infinite family of groups G(m,p,n), wherem,p,nare positive integers, and 34 +exceptional groups, denoted G4,G5,...,G 37by Shephard and Todd. +In this paper, we are only interested in finite complex reflection grou ps which are +well-generated . A complex reflection group of rank nis called well-generated if it is +generated by nreflections.1Well-generation can be equivalently characterised by a +duality property due to Orlik and Solomon [29]. Namely, a complex reflec tion group of +ranknhastwo sets ofdistinguished integers d1≤d2≤ ··· ≤dnandd∗ +1≥d∗ +2≥ ··· ≥d∗ +n, +called its degreesandcodegrees , respectively (see [24, p. 51 and Def. 10.27]). Orlik and +Solomon observed, using case-by-case checking, that an irreduc ible complex reflection +groupWof ranknis well-generated if and only if its degrees and codegrees satisfy +di+d∗ +i=dn +for alli= 1,2,...,n. The reader is referred to [24, App. D.2] for a table of the degree s +and codegrees of all irreducible complex reflection groups. Togeth er with the classi- +fication of Shephard and Todd [32], this constitutes a classification o f well-generated +complex reflection groups: the irreducible well-generated complex r eflection groups are +— the two infinite families G(d,1,n) andG(e,e,n), whered,e,nare positive inte- +gers, +— the exceptional groups G4,G5,G6,G8,G9,G10,G14,G16,G17,G18,G20,G21of +rank 2, +— the exceptional groups G23=H3,G24,G25,G26,G27of rank 3, +— the exceptional groups G28=F4,G29,G30=H4,G32of rank 4, +— the exceptional group G33of rank 5, +— the exceptional groups G34,G35=E6of rank 6, +— the exceptional group G36=E7of rank 7, +1We refer to [24, Def. 1.29] for the precise definition of “rank.” Roug hly speaking, the rank of a +complex reflection group Wis the minimal nsuch that Wcan be realized as reflection group on Cn.4 C. KRATTENTHALER AND T. W. M ¨ULLER +— and the exceptional group G37=E8of rank 8. +In this list, we have made visible the groups H3,F4,H4,E6,E7,E8which appear as +exceptional groups in the classification of all irreducible realreflection groups (cf. [16]). +LetWbe a well-generated complex reflection group of rank n, and letT⊆Wdenote +theset of all(complex) reflections inthegroup. Let ℓT:W→Zdenotethewordlength +in terms of the generators T. This word length is called absolute length orreflection +length. Furthermore, we define a partial order ≤TonWby +u≤Twif and only if ℓT(w) =ℓT(u)+ℓT(u−1w). (2.1) +This partial order is called absolute order orreflection order . As is well-known and +easy to see, the equation in (2.1) is equivalent to the statement tha t every shortest +representation of uby reflections occurs as an initial segment in some shortest produc t +representation of wby reflections. +Now fix a (generalised) Coxeter element2c∈Wand a positive integer m. The +m-divisible non-crossing partitions NCm(W) are defined as the set +NCm(W) =/braceleftbig +(w0;w1,...,w m) :w0w1···wm=cand +ℓT(w0)+ℓT(w1)+···+ℓT(wm) =ℓT(c)/bracerightbig +. +A partial order is defined on this set by +(w0;w1,...,w m)≤(u0;u1,...,u m) if and only if ui≤Twifor 1≤i≤m. +We have suppressed the dependence on c, since we understand this definition up to +isomorphism of posets. To be more precise, it can be shown that any two Coxeter +elements are related to each other by conjugation and (possibly) a n automorphism on +the field of complex numbers (see [33, Theorem 4.2] or [24, Cor. 11.2 5]), and hence the +resulting posets NCm(W) are isomorphic to each other. If m= 1, thenNC1(W) can +be identified with the set NC(W) of non-crossing partitions for the (complex) reflection +groupWasdefined byBessis andCorran(cf.[8]and[7, Sec.13]; theirdefinit ionextends +the earlier definition by Bessis [6] and Brady and Watt [10] for real r eflection groups). +The following result has been proved by a collaborative effort of seve ral authors (see +[7, Prop. 13.1]). +Theorem 1. LetWbe an irreducible well-generated complex reflection group, and let +d1≤d2≤ ··· ≤dnbe its degrees and h:=dnits Coxeter number. Then +|NCm(W)|=n/productdisplay +i=1mh+di +di. (2.2) +2An element of an irreducible well-generated complex reflection group Wof ranknis called a +Coxeter element if it isregularin the sense of Springer [33] (see also [24, Def. 11.21]) and of order dn. +An element of Wis called regular if it has an eigenvector which lies in no reflecting hyperp lane of a +reflection of W. It follows from an observation of Lehrer and Springer, proved un iformly by Lehrer +and Michel [23] (see [24, Theorem 11.28]), that there is always a regu lar element of order dnin an +irreducible well-generated complex reflection group Wof rankn. More generally, if a well-generated +complex reflection group Wdecomposes as W∼=W1×W2×···×Wk, where the Wi’s are irreducible, +then a Coxeter element of Wis an element of the form c=c1c2···ck, whereciis a Coxeter element of +Wi,i= 1,2,...,k. IfWis arealreflection group, that is, if all generators in Thave order 2, then the +notion of generalised Coxeter element given above reduces to that of a Coxeter element in the classical +sense (cf. [16, Sec. 3.16]).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 5 +Remark1.(1) The number in (2.2) is called the Fuß–Catalan number for the reflection +groupW. +(2) Ifcis a Coxeter element of a well-generated complex reflection group Wof rank +n, thenℓT(c) =n. (This follows from [7, Sec. 7].) +We conclude this section by recalling the definition of decomposition nu mbers from +[17, 18, 20]. Although we need them here only for (very small) real re flection groups, +and although, strictly speaking, they have been only defined for re al reflection groups in +[17, 18, 20], this definition can be extended to well-generated comple x reflection groups +without any extra effort, which we do now. +Given a well-generated complex reflection group Wof rankn, typesT1,T2,...,T d(in +the sense of the classification of well-generated complex reflection groups) such that the +sumoftheranksofthe Ti’sequalsn, andaCoxeter element c, thedecompositionnumber +NW(T1,T2,...,T d) is defined as the number of “minimal” factorisations c=c1c2···cd, +“minimal” meaning that ℓT(c1) +ℓT(c2) +···+ℓT(cd) =ℓT(c) =n, such that, for +i= 1,2,...,d, the type of cias a parabolic Coxeter element is Ti. (Here, the term +“parabolic Coxeter element” means a Coxeter element in some parab olic subgroup. It +follows from [31, Prop.6.3] that any element ciis indeed a Coxeter element in a unique +parabolic subgroup of W.3By definition, the type of ciis the type of this parabolic +subgroup.) Since any two Coxeter elements are related to each oth er by conjugation +plus field automorphism, the decomposition numbers are independen t of the choice of +the Coxeter element c. +The decomposition numbers for real reflection groups have been c omputed in [17, +18, 20]. To compute the decomposition numbers for well-generated complex reflection +groups is a task that remains to be done. +3.Cyclic sieving I +In this section we present the first cyclic sieving conjecture due to Armstrong [2, +Conj. 5.4.7], and to Bessis and Reiner [9, Conj. 6.4]. +Let��:NCm(W)→NCm(W) be the map defined by +(w0;w1,...,w m)/mapsto→/parenleftbig +(cwmc−1)w0(cwmc−1)−1;cwmc−1,w1,w2,...,w m−1/parenrightbig +.(3.1) +It is indeed not difficult to see that, if the ( m+ 1)-tuple on the left-hand side is an +element ofNCm(W), then so is the ( m+1)-tuple on the right-hand side. For m= 1, +this action reduces to conjugation by the Coxeter element c(applied to w1). Cyclic +sieving arising from conjugation by chas been the subject of [9]. +It is easy to see that φmhacts as the identity, where his the Coxeter number of W +(see (5.1) and Lemma 29 below). By slight abuse of notation, let C1be the cyclic group +of ordermhgenerated by φ. (The slight abuse consists in the fact that we insist on C1 +to be a cyclic group of order mh, while it may happen that the order of the action of +φgiven in (3.1) is actually a proper divisor of mh.) +3The uniqueness can be argued as follows: suppose that ciwere a Coxeter element in two parabolic +subgroups of W, sayU1andU2. Then it must also be a Coxeter element in the intersection U1∩U2. +On the other hand, the absolute length of a Coxeter element of a co mplex reflection group Uis always +equal to rk( U), the rank of U. (This follows from the fact that, for each element uofU, we have +ℓT(u) = codim/parenleftbig +ker(u−id)/parenrightbig +, with id denoting the identity element in U; see e.g. [31, Prop. 1.3]). We +conclude that ℓT(ci) = rk(U1) = rk(U2) = rk(U1∩U2), This implies that U1=U2.6 C. KRATTENTHALER AND T. W. M ¨ULLER +Given these definitions, we are now in the position to state the first c yclic sieving +conjecture of Armstrong, respectively of Bessis and Reiner. By t he results of [19] and +of this paper, it becomes the following theorem. +Theorem 2. For an irreducible well-generated complex reflection group Wand any +m≥1, the triple (NCm(W),Catm(W;q),C1), whereCatm(W;q)is theq-analogue of +the Fuß–Catalan number defined by +Catm(W;q) :=n/productdisplay +i=1[mh+di]q +[di]q, (3.2) +exhibits the cyclic sieving phenomenon in the sense of Reine r, Stanton and White [30]. +Here,nis the rank of W,d1,d2,...,d nare the degrees of W,his the Coxeter number +ofW, and[α]q:= (1−qα)/(1−q). +Remark2.We write Catm(W) for Catm(W;1). +By definition of the cyclic sieving phenomenon, we have to prove that Catm(W;q) is +a polynomial in qwith non-negative integer coefficients, and that +|FixNCm(W)(φp)|= Catm(W;q)/vextendsingle/vextendsingle +q=e2πip/mh, (3.3) +for allpin the range 0 ≤p0. +We begin with several auxiliary results.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 7 +Proposition 3. For all non-negative integers nandk, theq-binomial coefficient [n +k]q +is a polynomial in qwith non-negative integer coefficients. +Proof.This is a well-known fact, which can be derived either from the recurr ence rela- +tion(s) satisfied by the q-binomial coefficients (generalising Pascal’s recurrence relation +for binomial coefficients; cf. [1, eqs. (3.3.3) and (3.3.4)]), or from th e fact that the q- +binomial coefficient [n +k]qis the generating function for (integer) partitions with at most +kparts all of which are at most n−k(cf. [1, Theorem 3.1]). /square +Proposition 4. For all non-negative integers mandn, theq-Fuß–Catalan number of +typeAn, +1 +[(m+1)n+1]q/bracketleftbigg +(m+1)n+1 +n/bracketrightbigg +q, +is a polynomial in qwith non-negative integer coefficients. +Proof.In [25, Sec. 3.3], Loehr proves that +1 +[(m+1)n+1]q/bracketleftbigg +(m+1)n+1 +n/bracketrightbigg +q +=/summationdisplay +v∈V(m) +nqm(n +2)+/summationtext +i≥0(m(vi +2)−ivi)/productdisplay +i≥1qvi/summationtextm +j=1(m−j)vi−j/bracketleftbigg +vi+vi−1+···+vi−m−1 +vi/bracketrightbigg +q,(4.1) +whereV(m) +ndenotes the set of all sequences v= (v0,v1,...,v s) (for some s) of non- +negative integers with v0>0,vs>0, andv0+v1+···+vs=n, and such that there +is never a string of mor more consecutive zeroes in v. By convention, vi= 0 for all +negativei. His proof works by showing that the expressions on both sides of ( 4.1) +satisfy the same recurrence relation and initial conditions, using cla ssicalq-binomial +identities. We refer the reader to [25] for details. By Proposition 3, the expression on +the right-hand side of (4.1) is manifestly a polynomial in qwith non-negative integer +coefficients. /square +Lemma 5. Ifaandbare coprime positive integers, then +[ab]q +[a]q[b]q(4.2) +is a polynomial in qof degree (a−1)(b−1), all of whose coefficients are in {0,1,−1}. +Moreover, if one disregards the coefficients which are 0, then+1’s and(−1)’s alternate, +and the constant coefficient as well as the leading coefficient o f the polynomial equal +1. +Proof.LetΦn(q)denotethe n-thcyclotomicpolynomialin q. Usingtheclassicalformula +1−qn=/productdisplay +d|nΦd(q), +we see that +(1−q)(1−qab) +(1−qa)(1−qb)=/productdisplay +d1|a,d1/ne}ationslash=1 +d2|a,d2/ne}ationslash=1Φd1d2(q), +so that, manifestly, the expression in (4.2) is a polynomial in q. The claim concerning +the degree of this polynomial is obvious.8 C. KRATTENTHALER AND T. W. M ¨ULLER +In order to establish the claim on the coefficients, we start with a sub -expression of +(4.2), +(1−qab) +(1−qa)(1−qb)=/parenleftbiggb−1/summationdisplay +i=0qia/parenrightbigg/parenleftbigg∞/summationdisplay +j=0qjb/parenrightbigg +=∞/summationdisplay +k=0Ckqk, (4.3) +say. The assumption that aandbare coprime implies that 0 ≤Ck≤1 fork≤ +(a−1)(b−1). Multiplying both sides of (4.3) by 1 −q, we obtain the equation +[ab]q +[a]q[b]q= (1−q)(a−1)(b−1)/summationdisplay +k=0Ckqk+(1−q)∞/summationdisplay +k=(a−1)(b−1)+1Ckqk. (4.4) +By our previous observation on the coefficients Ckwithk≤(a−1)(b−1), it is obvious +that the coefficients of the first expression on the right-hand side of (4.4) are alternately ++1 and−1, when 0’s are disregarded. Since we already know that the left-ha nd side is +a polynomial in qof degree (a−1)(b−1), we may ignore the second expression. +The proof is concluded by observing that the claims on the constant and leading +coefficients are obvious. /square +Corollary 6. Letaandbbe coprime positive integers, and let γbe an integer with +γ≥(a−1)(b−1). Then the expression +[γ]q[ab]q +[a]q[b]q +is a polynomial in qwith non-negative integer coefficients. +Proof.Let +[ab]q +[a]q[b]q=(a−1)(b−1)/summationdisplay +k=0Dkqk. +We then have +[γ]q[ab]q +[a]q[b]q=(a−1)(b−1)+γ−1/summationdisplay +N=0qNN/summationdisplay +k=max{0,N−γ+1}Dk. (4.5) +IfN≤γ−1, then, by Lemma 5, the sum over kon the right-hand side of (4.5) equals +1−1+1−1+···, which is manifestly non-negative. On the other hand, if N >γ−1, +then we may rewrite the sum over kon the right-hand side of (4.5) as +N/summationdisplay +k=max{0,N−γ+1}Dk=(a−1)(b−1)/summationdisplay +k=N−γ+1Dk=(a−1)(b−1)+γ−1−N/summationdisplay +k=0D(a−1)(b−1)−k. +Again, by Lemma 5, this sum equals 1 −1 + 1−1 +···, which is manifestly non- +negative. /square +Lemma 7. Letαandβbe positive integers with α≥6andβ≥8. Then the expression +[α]q3[β]q4[72]q[3]q[4]q +[8]q[9]q[12]q +is a polynomial in qwith non-negative integer coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 9 +Proof.We have +[72]q[3]q[4]q +[8]q[9]q[12]q += (1−q3+q9−q15+q18)(1−q4+q8−q12+q16−q20+q24−q28+q32). +It should be observed that both factors on the right-hand side ha ve the property that +coefficients are in {0,1,−1}and that (+1)’s and ( −1)’s alternate, if one disregards the +coefficients which are 0. If we now apply the same idea as in the proof o f Corollary 6, +then we see that [ α]q3times the first factor is a polynomial in qwith non-negative +integer coefficients, as is [ β]q4times the second factor. Taken together, this establishes +the claim. /square +Lemma 8. Letαandβbe positive integers with α≥26andβ≥8. Then the expression +[α]q[β]q4[15]q +[3]q[5]q[72]q[3]q[4]q +[8]q[9]q[12]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[15]q[72]q[4]q +[5]q[8]q[9]q[12]q += (1−q+q5−q6+q9−q11+q12−q13+q14−q15+q17−q20+q21−q25+q26) +×(1−q4+q8−q12+q16−q20+q24−q28+q32). +Again, if we apply the same idea as in the proof of Corollary 6, then we s ee that [α]q +times the first factor on the the right-hand side of the above equa tion is a polynomial +inqwith non-negative integer coefficients, as is [ β]q4times the second factor. Taken +together, this establishes the claim. /square +Lemma 9. Letαandβbe positive integers with α≥18andβ≥3. Then the expression +[α]q3[β]q4[90]q[3]q[4]q +[5]q[6]q[9]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[90]q[3]q[4]q +[5]q[6]q[9]q += (1−q3+q9−q12+q18−q21+q27−q33+q36−q42+q45−q51+q54) +×(1−q4+q5+q6−q9+q11+q12−q14+q17+q18−q19+q23) += (1−q3+q9−q12+q18−q21+q27−q33+q36−q42+q45−q51+q54) +×/parenleftbig +1−q4+q12+q5(1−q4+q12)+q6(1−q8+q12)+q11(1−q8+q12)/parenrightbig +. +Again, if we apply the same idea as in the proof of Corollary 6, then we s ee that [α]q3 +times the first factor on the the right-hand side of the above equa tion is a polynomial +inqwith non-negative integer coefficients, as is [ β]q4times the second factor. Taken +together, this establishes the claim. /square10 C. KRATTENTHALER AND T. W. M ¨ULLER +Lemma 10. Letαandβbe positive integers with α≥20andβ≥18. Then the +expression +[α]q[β]q3[90]q[3]q +[5]q[6]q[9]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[90]q[3]q +[5]q[6]q[9]q += (1−q3+q9−q12+q18−q21+q27−q33+q36−q42+q45−q51+q54) +×(1−q+q5−q7+q10−q13+q15−q19+q20). +Again, if we apply the same idea as in the proof of Corollary 6, then we s ee that +[α]q3times the second factor on the the right-hand side of the above eq uation is a +polynomial in qwith non-negative integer coefficients, as is [ β]q3times the first factor. +Taken together, this establishes the claim. /square +Lemma 11. Letαbe a positive integer with α≥26. Then the expression +[α]q[15]q +[3]q[5]q[12]q3 +[3]q3[4]q3 +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[15]q +[3]q[5]q[12]q3 +[3]q3[4]q3 += 1−q+q5−q6+q9−q11+q12−q13+q14−q15+q17−q20+q21−q25+q26. +Once again, the coefficients of the polynomial on the right-hand side are in{0,1,−1} +and (+1)’s and ( −1)’s alternate, if one disregards the coefficients which are 0. The +argument from the proof of Corollary 6 then implies that this is a polyn omial inqwith +non-negative integer coefficients. /square +Lemma 12. Letαbe a positive integer with α≥14. Then the expression +[α]q[15]q +[3]q[5]q[6]q3 +[2]q3[3]q3 +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[15]q +[3]q[5]q[6]q3 +[2]q3[3]q3= 1−q+q5−q7+q9−q13+q14. +Repeating the arguments of the previous proof, this establishes t he claim. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11 +Lemma 13. Letαandβbe positive integers with α≥30andβ≥20. Then the +expression +[α]q[β]q2[84]q[2]q +[4]q[6]q[7]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[84]q[2]q +[4]q[6]q[7]q += (1−q+q6−q8+q12−q15+q18−q22+q24−q29+q30) +×(1−q2+q4−q6+q8−q10+q12−q14+q16−q18+q20−q22 ++q24−q26+q28−q30+q32−q34+q36−q38+q40). +Onceagain, ifweapplythesameideaasintheproofofCorollary6, the nweseethat[ α]q +times the first factor on the the right-hand side of the above equa tion is a polynomial +inqwith non-negative integer coefficients, as is [ β]q2times the second factor. Taken +together, this establishes the claim. /square +Lemma 14. Letαandβbe positive integers with α≥24andβ≥68. Then the +expression +[α]q[β]q[105]q +[3]q[5]q[7]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[105]q +[3]q[5]q[7]q += (1−q+q5−q6+q7−q8+q10−q11+q12−q13+q14−q16 ++q17−q18+q19−q23+q24) +×(1−q+q3−q4+q6−q7+q9−q10+q12−q13+q15−q16+q18−q19 ++q21−q22+q24−q25+q27−q28+q30−q31+q33−q34+q35−q37 ++q38−q40+q41−q43+q44−q46+q47−q49+q50−q52+q53−q55 ++q56−q58+q59−q61+q62−q64+q65−q67+q68). +Once again, if we apply the same idea as in the proof of Corollary 6, the n we see +that [α]qtimes the first factor on the the right-hand side of the above equa tion is a +polynomial in qwith non-negative integer coefficients, as is [ β]qtimes the second factor. +Taken together, this establishes the claim. /square +Lemma 15. Letαandβbe positive integers with α≥24andβ≥34. Then the +expression +[α]q[β]q[70]q +[2]q[5]q[7]q +is a polynomial in qwith non-negative integer coefficients.12 C. KRATTENTHALER AND T. W. M ¨ULLER +Proof.We have +[70]q +[2]q[5]q[7]q += (1−q+q5−q6+q7−q8+q10−q11+q12−q13 ++q14−q16+q17−q18+q19−q23+q24) +×(1−q+q2−q3+q4−q5+q6−q7+q8−q9+q10−q11+q12−q13 ++q14−q15+q16−q17+q18−q19+q20−q21+q22−q23+q24−q25 ++q26−q27+q28−q29+q30−q31+q32−q33+q34). +Also here, if we apply the same idea as in the proof of Corollary 6, then we see that [ α]q +times the first factor on the the right-hand side of the above equa tion is a polynomial +inqwith non-negative integer coefficients, as is [ β]qtimes the second factor. Taken +together, this establishes the claim. /square +Lemma 16. Letαandβbe positive integers with α≥4andβ≥2. Then the expression +[α]q2[β]q5[30]q[2]q[3]q[5]q +[6]q[10]q[15]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[30]q[2]q[3]q[5]q +[6]q[10]q[15]q= 1+q−q3−q4−q5+q7+q8. +If we multiply this expression by [ α]q2, then, forα= 4 we obtain +1+q+q2−q5−q9+q12+q13+q14, +forα= 5 we obtain +1+q+q2−q5+q8−q11+q14+q15+q16, +and, forα≥6, we obtain +1+q+q2−q5+q8+q10+p1(q)+q2α−4+q2α−2−q2α+1+q2α+4+q2α+5+q2α+6, +wherep1(q) is a polynomial in qwith non-negative coefficients of order at least 11 and +degree at most 2 α−5. In all cases it is obvious that the product of the result and [ β]q5, +withβ≥2, is a polynomial in qwith non-negative coefficients. /square +Lemma 17. Letαandβbe positive integers with α≥14andβ≥2. Then the +expression +[α]q[β]q5[14]q +[2]q[7]q[30]q[2]q[3]q[5]q +[6]q[10]q[15]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[14]q +[2]q[7]q[30]q[2]q[3]q[5]q +[6]q[10]q[15]q= 1−q3−q5+q6+q7+q8−q9−q11+q14.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13 +If we multiply this expression by [ α]q, then, forα≥14, we obtain +1+q+q2−q5+q7+2q8+q9+q10+p2(q) ++qα+3+qα+4+2qα+5+qα+6−qα+8+qα+11+qα+12+qα+13, +wherep2(q) is a polynomial in qwith non-negative coefficients of order at least 11 and +degree at most α+2. It is obvious that the product of the result and [ β]q5, withβ≥2, +is a polynomial in qwith non-negative coefficients. /square +Lemma 18. Letαandβbe positive integers with α≥32andβ≥12. Then the +expression +[α]q[β]q2[35]q +[5]q[7]q[30]q[2]q[3]q[5]q +[6]q[10]q[15]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[35]q +[5]q[7]q[30]q[2]q[3]q[5]q +[6]q[10]q[15]q= 1−q2−q3+q5+q6+q7−q8−2q9+q11+q12−q16+q20+q21 +−2q23−q24+q25+q26+q27−q29−q30+q32. +If we multiply this expression by [ α]q, then, forα≥32, we obtain +1+q−q3−q4+q6+2q7+q8−q9−q10+q12+q13+q14+q15+q20+2q21+2q22 +−q24+q26+2q27+2q28+q29+p3(q)+qα+2+2qα+4+2qα+5+qα+6−qα+8 ++2qα+9+2qα+10+qα+11+qα+16+qα+17+qα+18+qα+19−qα+21−qα+22+qα+23 ++2qα+24+qα+25−qα+27−qα+28+qα+30+qα+31, +wherep3(q) is a polynomial in qwith non-negative coefficients of order at least 30 and +degree at most α+1. It is obvious that the product of the result and [ β]q2, withβ≥12, +is a polynomial in qwith non-negative coefficients. /square +Lemma 19. Letαandβbe positive integers with α≥16andβ≥2. Then the +expression +[α]q2[β]q5[60]q[2]q[3]q[5]q +[10]q[12]q[15]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[60]q[2]q[3]q[5]q +[10]q[12]q[15]q= 1+q−q3−q4−q5−q6+q8+q9+q10+q11+q12−q14−q15−q16 +−q17−q18+q20+q21+q22+q23+q24−q26−q27−q28−q29+q31+q32. +If we multiply this expression by [ α]q2, then, forα≥16, we obtain +1+q+q2−q5−q6−q7+q10+q11+2q12+q13+q14 +−q17−q18−q19+q22+q23+2q24+p4(q)+s4(q),14 C. KRATTENTHALER AND T. W. M ¨ULLER +wherep4(q) is a polynomial in qwith non-negative coefficients of order at least 25 +and degree at most 2 α+ 5 ands4(q) completes the above expression to a symmetric +polynomial. It is obvious that the product of the result and [ β]q5, withβ≥2, is a +polynomial in qwith non-negative coefficients. /square +Lemma 20. Letαandβbe positive integers with α≥56andβ≥4. Then the +expression +[α]q[β]q2[35]q +[5]q[7]q[60]q[2]q[3]q[5]q +[10]q[12]q[15]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[35]q +[5]q[7]q[60]q[2]q[3]q[5]q +[10]q[12]q[15]q= 1−q2−q3+q5+q7−q9+q12−q13+q15−q16−q18 ++q19+q20+q22−2q23+q27−q28+q29−2q33+q34+q36+q37−q38 +−q40+q41−q43+q44−q47+q49+q51−q53−q54+q56. +If we multiply this expression by [ α]q, then, forα≥56, we obtain +1+q−q3−q4+q7+q8+q12+q15−q18+q20+q21+2q22+q27 ++q29+q30+q31+q32−q33+q36+2q37+q38+p5(q)+s5(q), +wherep5(q) is a polynomial in qwith non-negative coefficients of order at least 39 +and degree at most α+ 16 ands5(q) completes the above expression to a symmetric +polynomial. It is obvious that the product of the result and [ β]q2, withβ≥4, is a +polynomial in qwith non-negative coefficients. /square +Lemma 21. Letαandβbe positive integers with α≥38andβ≥2. Then the +expression +[α]q[β]q5[14]q +[2]q[7]q[60]q[2]q[3]q[5]q +[10]q[12]q[15]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[14]q +[2]q[7]q[60]q[2]q[3]q[5]q +[10]q[12]q[15]q= 1−q3−q5+q7+q8−q13+q19−q25+q30+q31−q33−q35+q38. +If we multiply this expression by [ α]q, then, forα≥38, we obtain +1+q+q2−q5−q6+q8+q9+q10+q11+p6(q)+s6(q), +wherep6(q) is a polynomial in qwith non-negative coefficients of order at least 12 +and degree at most α+ 25 ands6(q) completes the above expression to a symmetric +polynomial. It is obvious that the product of the result and [ β]q2, withβ≥2, is a +polynomial in qwith non-negative coefficients. /square +Lemma 22. Letαandβbe positive integers with α≥30andβ≥26. Then the +expression +[α]q[β]q3[126]q[3]q +[6]q[7]q[9]qCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15 +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[126]q[3]q +[6]q[7]q[9]q= (1−q+q6−q8+q12−q15+q18−q22+q24−q29+q30) +×(1−q3+q9−q12+q18−q21+q27−q30+q36−q39 ++q42−q48+q51−q57+q60−q66+q69−q75+q78). +If we apply the same idea as in the proof of Corollary 6, then we see th at [α]qtimes the +first factor on the the right-hand side of the above equation is a po lynomial in qwith +non-negative integer coefficients, as is [ β]q3times the second factor. Taken together, +this establishes the claim. /square +Lemma 23. Letαandβbe positive integers with α≥66andβ≥54. Then the +expression +[α]q[β]q3[252]q[3]q +[7]q[9]q[12]q +is a polynomial in qwith non-negative integer coefficients. +Proof.We have +[252]q[3]q +[7]q[9]q[12]q += (1−q+q7−q8+q12−q13+q14−q15+q19−q20+q21−q22+q24−q25 ++q26−q27+q28−q29+q31−q32+q33−q34+q35−q37+q38 +−q39+q40−q41+q42−q44+q45−q46+q47−q51+q52−q53 ++q54−q58+q59−q65+q66) +×(1−q3+q9−q12+q18−q21+q27−q30+q36−q39+q45−q48 ++q54−q57+q63−q66+q72−q75+q81−q87+q90−q96+q99 +−q105+q108−q114+q117−q123+q126−q132+q135−q141 ++q144−q150+q153−q159+q162). +If we apply the same idea as in the proof of Corollary 6, then we see th at [α]qtimes the +first factor on the the right-hand side of the above equation is a po lynomial in qwith +non-negative integer coefficients, as is [ β]q3times the second factor. Taken together, +this establishes the claim. /square +Lemma 24. Letαandβbe positive integers with α≥54andβ≥34. Then the +expression +[α]q[β]q2[140]q[2]q +[4]q[7]q[10]q +is a polynomial in qwith non-negative integer coefficients.16 C. KRATTENTHALER AND T. W. M ¨ULLER +Proof.We have +[140]q[2]q +[4]q[7]q[10]q += (1−q+q7−q8+q10−q11+q14−q15+q17−q18+q20−q22+q24−q25 ++q27−q29+q30−q32+q34−q36+q37−q39+q40−q43+q44−q46 ++q47−q53+q54) +×(1−q2+q4−q6+q8−q10+q12−q14+q16−q18+q20−q22+q24 +−q26+q28−q30+q32−q34+q36−q38+q40−q42+q44−q46 ++q48−q50+q52−q54+q56−q58+q60−q62+q64−q66+q68), +If we apply the same idea as in the proof of Corollary 6, then we see th at [α]qtimes the +first factor on the the right-hand side of the above equation is a po lynomial in qwith +non-negative integer coefficients, as is [ β]q2times the second factor. Taken together, +this establishes the claim. /square +We are now ready for the proof of the main result of this section. +Theorem 25. For all irreducible well-generated complex reflection grou ps and posi- +tive integers m, theq-Fuß–Catalan number Catm(W;q)is a polynomial in qwith non- +negative integer coefficients. +Proof.First, letW=An. In this case, the degrees are 2 ,3,...,n+1, and hence +Catm(An;q) =1 +[(m+1)n+1]q/bracketleftbigg +(m+1)n+1 +n/bracketrightbigg +q, +which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients. +Next, letW=G(d,1,n). In this case, the degrees are d,2d,...,nd , and hence +Catm(G(d,1,n);q) =/bracketleftbigg +(m+1)n +n/bracketrightbigg +qd, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients. +Now, letW=G(e,e,n). In this case, the degrees are e,2e,...,(n−1)e,n, and hence +Catm(G(e,e,n);q) =[m(n−1)e+n]q +[n]qn−1/productdisplay +i=1[m(n−1)e+ie]q +[ie]q +=/bracketleftbigg +(m+1)(n−1) +n−1/bracketrightbigg +qe+qn[e]qn/bracketleftbigg +(m+1)(n−1) +n/bracketrightbigg +qe, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients. +It remains to verify the claim for the exceptional groups. +ForW=G4, the degrees are 4 ,6, and hence +Catm(G4;q) =[6m+4]q[6m+6]q +[4]q[6]q=1 +[3m+4]q2/bracketleftbigg +3m+4 +3/bracketrightbigg +q2, +which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 17 +ForW=G5, the degrees are 6 ,12, and hence +Catm(G5;q) =[12m+6]q[12m+12]q +[6]q[12]q=/bracketleftbigg +2m+2 +2/bracketrightbigg +q6, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients. +ForW=G6, the degrees are 4 ,12, and hence +Catm(G6;q) =[12m+4]q[12m+12]q +[4]q[12]q= [3m+1]q4[m+1]q12, +which is manifestly a polynomial in qwith non-negative integer coefficients. +ForW=G8, the degrees are 8 ,12, and hence +Catm(G8;q) =[12m+8]q[12m+12]q +[8]q[12]q=1 +[3m+4]q4/bracketleftbigg +3m+4 +3/bracketrightbigg +q4, +which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients. +ForW=G9, the degrees are 8 ,24, and hence +Catm(G9;q) =[24m+8]q[24m+24]q +[8]q[24]q= [3m+1]q8[m+1]q24, +which is manifestly a polynomial in qwith non-negative integer coefficients. +ForW=G10, the degrees are 12 ,24, and hence +Catm(G10;q) =[24m+12]q[24m+24]q +[12]q[24]q=/bracketleftbigg +2m+2 +2/bracketrightbigg +q12, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients. +ForW=G14, the degrees are 6 ,24, and hence +Catm(G14;q) =[24m+6]q[24m+24]q +[6]q[24]q= [4m+1]q6[m+1]q24, +which is manifestly a polynomial in qwith non-negative integer coefficients. +ForW=G16, the degrees are 20 ,30, and hence +Catm(G16;q) =[30m+20]q[30m+30]q +[20]q[30]q=1 +[3m+4]q10/bracketleftbigg +3m+4 +3/bracketrightbigg +q10, +which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients. +ForW=G17, the degrees are 20 ,60, and hence +Catm(G17;q) =[60m+20]q[60m+60]q +[20]q[60]q= [3m+1]q20[m+1]q60, +which is manifestly a polynomial in qwith non-negative integer coefficients. +ForW=G18, the degrees are 30 ,60, and hence +Catm(G18;q) =[60m+30]q[60m+60]q +[30]q[60]q=/bracketleftbigg +2m+2 +2/bracketrightbigg +q30, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.18 C. KRATTENTHALER AND T. W. M ¨ULLER +ForW=G20, the degrees are 12 ,30, and hence +Catm(G20;q) =[30m+12]q[30m+30]q +[12]q[30]q +=/braceleftBigg/bracketleftbig5m+2 +2/bracketrightbig +q12[m+1]q30, ifmis even, +[5m+2]q6/bracketleftbigm+1 +2/bracketrightbig +q60[10]q6 +[2]q6[5]q6,ifmis odd, +which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in +both cases. +ForW=G21, the degrees are 12 ,60, and hence +Catm(G21;q) =[60m+12]q[60m+60]q +[12]q[60]q= [5m+1]q12[m+1]q60, +which is manifestly a polynomial in qwith non-negative integer coefficients. +ForW=G23=H3, the degrees are 2 ,6,10, and hence +Catm(H3;q) =[10m+2]q[10m+6]q[10m+10]q +[2]q[6]q[10]q += + +[5m+2]q2/bracketleftbig5m+3 +3/bracketrightbig +q6[m+1]q10, ifm≡0 (mod 3),/bracketleftbig5m+1 +3/bracketrightbig +q6[5m+3]q2[m+1]q10, ifm≡1 (mod 3), +[5m+2]q2[5m+3]q2/bracketleftbigm+1 +3/bracketrightbig +q30[15]q2 +[3]q2[5]q2,ifm≡2 (mod 3), +which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in all +cases. +ForW=G24, the degrees are 4 ,6,14, and hence +Catm(G24;q) =[14m+4]q[14m+6]q[14m+14]q +[4]q[6]q[14]q. +We have +Catm(G24;q) = + +/bracketleftbig7m +2+1/bracketrightbig +q4/bracketleftbig14m +6+1/bracketrightbig +q6[m+1]q14,ifm≡0 (mod 6),/bracketleftbig7m+2 +3/bracketrightbig +q6/bracketleftbig7m+3 +2/bracketrightbig +q4[m+1]q14, ifm≡1 (mod 6), +/bracketleftbig7m +2+1/bracketrightbig +q4[7m+3]q2/bracketleftbigm+1 +3/bracketrightbig +q42[21]q2 +[3]q2[7]q2,ifm≡2 (mod 6), +[7m+2]q2/bracketleftbig7m +3+1/bracketrightbig +q6/bracketleftbigm+1 +2/bracketrightbig +q28[14]q2 +[2]q2[7]q2,ifm≡3 (mod 6), +/bracketleftbig7m+2 +6/bracketrightbig +q12[6]q2 +[2]q2[3]q2[7m+3]q2[m+1]q14,ifm≡4 (mod 6), +[7m+2]q2/bracketleftbig7m+3 +2/bracketrightbig +q4/bracketleftbigm+1 +3/bracketrightbig +q42[21]q2 +[3]q2[7]q2,ifm≡5 (mod 6), +which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in all +cases. +ForW=G25, the degrees are 6 ,9,12, and hence +Catm(G25;q) =[12m+6]q[12m+9]q[12m+12]q +[6]q[9]q[12]q=1 +[4m+5]q3/bracketleftbigg +4m+5 +4/bracketrightbigg +q3, +which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 19 +ForW=G26, the degrees are 6 ,12,18, and hence +Catm(G26;q) =[18m+6]q[18m+12]q[18m+18]q +[6]q[12]q[18]q=/bracketleftbigg +3m+3 +3/bracketrightbigg +q6, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients. +ForW=G27, the degrees are 6 ,12,30, and hence +Catm(G27;q) =[30m+6]q[30m+12]q[30m+30]q +[6]q[12]q[30]q= [m+1]q30/bracketleftbigg +5m+2 +2/bracketrightbigg +q6, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients. +ForW=G28=F4, the degrees are 2 ,6,8,12, and hence +Catm(F4;q) =[12m+2]q[12m+6]q[12m+8]q[12m+12]q +[2]q[6]q[8]q[12]q += + +[6m+1]q2[2m+1]q6/bracketleftbig3 +2m+1/bracketrightbig +q8[m+1]q12, ifmis even, +[6m+1]q2[2m+1]q6/bracketleftbigm+1 +2/bracketrightbig +q24[3m+2]q4[6]q4 +[2]q4[3]q4,ifmis odd, +which in both cases is a polynomial in qwith non-negative integer coefficients; in the +second case this is due to Corollary 6. +ForW=G29, the degrees are 4 ,8,12,20, and hence +Catm(G29;q) =[20m+4]q[20m+8]q[20m+12]q[20m+20]q +[4]q[8]q[12]q[20]q += [m+1]q20/bracketleftbigg +5m+3 +3/bracketrightbigg +q4, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients. +ForW=G30=H4, the degrees are 2 ,12,20,30, and hence +Catm(H4;q) =[30m+2]q[30m+12]q[30m+20]q[30m+30]q +[2]q[12]q[20]q[30]q. +Ifmis even, then we have +Catm(H4;q) = [15m+1]q2/bracketleftbig5 +2m+1/bracketrightbig +q12/bracketleftbig3 +2m+1/bracketrightbig +q20[m+1]q30, +which is manifestly a polynomial in qwith non-negative integer coefficients. On the +other hand, if mis odd, then we may write +Catm(H4;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[5m+2]q6[3m+2]q10/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q6[10]q2[15]q2, +which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients. +ForW=G32, the degrees are 12 ,18,24,30, and hence +Catm(G32;q) =[30m+12]q[30m+18]q[30m+24]q[30m+30]q +[12]q[18]q[24]q[30]q +=1 +[5m+6]q6/bracketleftbigg +5m+6 +5/bracketrightbigg +q6, +which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.20 C. KRATTENTHALER AND T. W. M ¨ULLER +ForW=G33, the degrees are 4 ,6,10,12,18, and hence +Catm(G33;q) =[18m+4]q[18m+6]q[18m+10]q[18m+12]q[18m+18]q +[4]q[6]q[10]q[12]q[18]q. +Ifm≡0 (mod 10), then we have +Catm(G33;q) =/bracketleftbig9 +2m+1/bracketrightbig +q4[3m+1]q6/bracketleftbig9 +5m+1/bracketrightbig +q10/bracketleftbig3 +2m+1/bracketrightbig +q12[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +1 (mod 10), then we have +Catm(G33;q) =/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+5 +2/bracketrightbig +q4/bracketleftbig3m+2 +5/bracketrightbig +q30[m+1]q18[9m+2]q2[15]q2 +[3]q2[5]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡2 (mod 10), then we have +Catm(G33;q) =/bracketleftbig9m+2 +10/bracketrightbig +q20[3m+1]q6/bracketleftbig3 +2m+1/bracketrightbig +q12[m+1]q18[9m+5]q2[10]q2 +[2]q2[5]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡3 (mod 10), then we have +Catm(G33;q) =/bracketleftbig3m+1 +10/bracketrightbig +q60/bracketleftbig9m+5 +2/bracketrightbig +q4[3m+2]q6[m+1]q18[9m+2]q2[30]q2 +[5]q2[6]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡4 (mod 10), then we have +Catm(G33;q) =/bracketleftbig9 +2m+1/bracketrightbig +q4[3m+1]q6/bracketleftbig3 +2m+1/bracketrightbig +q12/bracketleftbigm+1 +5/bracketrightbig +q90[9m+5]q2[45]q2 +[5]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡5 (mod 10), then we have +Catm(G33;q) =/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+5 +10/bracketrightbig +q20[3m+2]q6[m+1]q18[9m+2]q2[10]q2 +[2]q2[5]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡6 (mod 10), then we have +Catm(G33;q) =/bracketleftbig9 +2m+1/bracketrightbig +q4[3m+1]q6/bracketleftbig3m+2 +10/bracketrightbig +q60[m+1]q18[9m+5]q2[30]q2 +[5]q2[6]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡7 (mod 10), then we have +Catm(G33;q) =/bracketleftbig9m+2 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+5 +2/bracketrightbig +q4[3m+2]q6[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +8 (mod 10), then we have +Catm(G33;q) =/bracketleftbig9 +2m+1/bracketrightbig +q4/bracketleftbig3m+1 +5/bracketrightbig +q30/bracketleftbig3 +2m+1/bracketrightbig +q12[m+1]q18[9m+5]q2[15]q2 +[3]q2[5]q2,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 21 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. Fi- +nally, ifm≡9 (mod 10), then we have +Catm(G33;q) =/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+5 +2/bracketrightbig +q4[3m+2]q6/bracketleftbigm+1 +5/bracketrightbig +q90[9m+2]q2[45]q2 +[5]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. +ForW=G34, the degrees are 6 ,12,18,24,30,42, and hence +Catm(G34;q) =[42m+6]q[42m+12]q[42m+18]q[42m+24]q[42m+30]q[42m+42]q +[6]q[12]q[18]q[24]q[30]q[42]q += [m+1]q42/bracketleftbigg +7m+5 +5/bracketrightbigg +q6, +which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients. +ForW=G35=E6, the degrees are 2 ,5,6,8,9,12, and hence +Catm(E6;q) =[12m+2]q[12m+5]q[12m+6]q[12m+8]q[12m+9]q[12m+12]q +[2]q[5]q[6]q[8]q[9]q[12]q. +Ifm≡0 (mod 30),then we have +Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5 +5/bracketrightbig +q5[2m+1]q6/bracketleftbig3m+2 +2/bracketrightbig +q8/bracketleftbig4m+3 +3/bracketrightbig +q9[m+1]q12, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +1 (mod 30),then we have +Catm(E6;q) = [6m+1]q2/bracketleftbig2m+1 +3/bracketrightbig +q18[4m+3]q3[6]q3 +[2]q3[3]q3 +×/bracketleftbig3m+2 +5/bracketrightbig +q20[12m+5]q[20]q +[4]q[5]q/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4. +If one decomposes [6 m+1]q2as [3m+1]q4+q2[3m]q4, then one sees that, by Corollary 6, +the above expression is a polynomial in qwith non-negative integer coefficients. If +m≡2 (mod 30),then we have +Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1 +5/bracketrightbig +q30[30]q +[5]q[6]q +×/bracketleftbig3m+2 +2/bracketrightbig +q8[4m+3]q3/bracketleftbigm+1 +3/bracketrightbig +q36[12]q3 +[3]q3[4]q3, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡3 (mod 30),then we have +Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6[3m+2]q4 +×/bracketleftbig4m+3 +15/bracketrightbig +q45[45]q +[5]q[9]q/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4,22 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡4 (mod 30),then we have +Catm(E6;q) =/bracketleftbig12m+2 +5/bracketrightbig +q5/bracketleftbig12m+5 +2/bracketrightbig +q2/bracketleftbig2m+1 +3/bracketrightbig +q18[6]q3 +[2]q3[3]q3 +×/bracketleftbig3m+2 +2/bracketrightbig +q8[4m+3]q3[m+1]q12, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡5 (mod 30),then we have +Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5 +5/bracketrightbig +q5[2m+1]q6 +×[3m+2]q4[4m+3]q3/bracketleftbigm+1 +6/bracketrightbig +q72[72]q[3]q[4]q +[8]q[9]q[12]q, +which, by Lemma 7, is a polynomial in qwith non-negative integer coefficients. If +m≡6 (mod 30),then we have +Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2 +10/bracketrightbig +q40[40]q +[5]q[8]q/bracketleftbig4m+3 +3/bracketrightbig +q9[m+1]q12, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡7 (mod 30),then we have +Catm(E6;q) =/bracketleftbig6m+1 +2/bracketrightbig +q4[12m+5]q/bracketleftbig2m+1 +15/bracketrightbig +q90 +×[90]q[3]q[4]q +[5]q[6]q[9]q[3m+2]q4[4m+3]q3/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4, +which, by Corollary 6 and Lemma 9, is a polynomial in qwith non-negative integer +coefficients. If m≡8 (mod 30),then we have +Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2 +2/bracketrightbig +q8 +×/bracketleftbig4m+3 +5/bracketrightbig +q15[15]q +[3]q[5]q/bracketleftbigm+1 +3/bracketrightbig +q36[12]q3 +[3]q3[4]q3, +which, by Lemma 11, is a polynomial in qwith non-negative integer coefficients. If +m≡9 (mod 30),then we have +Catm(E6;q) =/bracketleftbig12m+2 +5/bracketrightbig +q5/bracketleftbig12m+5 +2/bracketrightbig +q2[2m+1]q6[3m+2]q4/bracketleftbig4m+3 +3/bracketrightbig +q9/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡10 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5 +5/bracketrightbig +q5/bracketleftbig2m+1 +3/bracketrightbig +q18[6]q3 +[2]q3[3]q3/bracketleftbig3m+2 +2/bracketrightbig +q8[4m+3]q3[m+1]q12,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 23 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡11 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6 +×/bracketleftbig3m+2 +5/bracketrightbig +q20[20]q +[4]q[5]q[4m+3]q3/bracketleftbigm+1 +6/bracketrightbig +q72[72]q[3]q[4]q +[8]q[9]q[12]q. +If one decomposes [6 m+1]q2as [3m+1]q4+q2[3m]q4, then one sees that, by Corollary 6 +and Lemma 7, this is a polynomial in qwith non-negative integer coefficients. If m≡ +12 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1 +5/bracketrightbig +q30[30]q +[5]q[6]q/bracketleftbig3m+2 +2/bracketrightbig +q8/bracketleftbig4m+3 +3/bracketrightbig +q9[m+1]q12, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡13 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1 +3/bracketrightbig +q18[6]q3 +[2]q3[3]q3 +×[3m+2]q4/bracketleftbig4m+3 +5/bracketrightbig +q15[15]q +[3]q[5]q/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4, +which, by Lemma 12, is a polynomial in qwith non-negative integer coefficients. If +m≡14 (mod 30) ,then we have +Catm(E6;q) =/bracketleftbig12m+2 +5/bracketrightbig +q5/bracketleftbig12m+5 +2/bracketrightbig +q2[2m+1]q6/bracketleftbig3m+2 +2/bracketrightbig +q8[4m+3]q3/bracketleftbigm+1 +3/bracketrightbig +q36[12]q3 +[3]q3[4]q3, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡15 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5 +5/bracketrightbig +q5[2m+1]q6[3m+2]q4/bracketleftbig4m+3 +3/bracketrightbig +q9/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡16 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1 +3/bracketrightbig +q18[6]q3 +[2]q3[3]q3 +×/bracketleftbig3m+2 +10/bracketrightbig +q40[40]q +[5]q[8]q[4m+3]q3[m+1]q12, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡17 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1 +5/bracketrightbig +q30[30]q +[5]q[6]q +×[3m+2]q4[4m+3]q3/bracketleftbigm+1 +6/bracketrightbig +q72[72]q[3]q[4]q +[8]q[9]q[12]q,24 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6 and Lemma 7, is a polynomial in qwith non-negative integer +coefficients. If m≡18 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2 +2/bracketrightbig +q8/bracketleftbig4m+3 +15/bracketrightbig +q45[45]q +[5]q[9]q[m+1]q12, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡19 (mod 30) ,then we have +Catm(E6;q) =/bracketleftbig12m+2 +5/bracketrightbig +q5/bracketleftbig12m+5 +2/bracketrightbig +q2/bracketleftbig2m+1 +3/bracketrightbig +q18[6]q3 +[2]q3[3]q3 +×[3m+2]q4[4m+3]q3/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡20 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5 +5/bracketrightbig +q5[2m+1]q6/bracketleftbig3m+2 +2/bracketrightbig +q8[4m+3]q3/bracketleftbigm+1 +3/bracketrightbig +q36[12]q3 +[3]q3[4]q3, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡21 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6 +×/bracketleftbig3m+2 +5/bracketrightbig +q20[20]q +[4]q[5]q/bracketleftbig4m+3 +3/bracketrightbig +q9/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡22 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1 +15/bracketrightbig +q90[90]q[3]q +[5]q[6]q[9]q +×/bracketleftbig3m+2 +2/bracketrightbig +q8[4m+3]q3[m+1]q12, +which, by Lemma 10, is a polynomial in qwith non-negative integer coefficients. If +m≡23 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6 +×[3m+2]q4/bracketleftbig4m+3 +5/bracketrightbig +q15[15]q +[3]q[5]q/bracketleftbigm+1 +6/bracketrightbig +q72[72]q[3]q[4]q +[8]q[9]q[12]q, +which, by Lemma 8, is a polynomial in qwith non-negative integer coefficients. If +m≡24 (mod 30) ,then we have +Catm(E6;q) =/bracketleftbig12m+2 +5/bracketrightbig +q5/bracketleftbig12m+5 +2/bracketrightbig +q2[2m+1]q6/bracketleftbig3m+2 +2/bracketrightbig +q8/bracketleftbig4m+3 +3/bracketrightbig +q9[m+1]q12,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 25 +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +25 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5 +5/bracketrightbig +q5/bracketleftbig2m+1 +3/bracketrightbig +q18[6]q3 +[2]q3[3]q3 +×[3m+2]q4[4m+3]q3/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡26 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6 +×/bracketleftbig3m+2 +10/bracketrightbig +q40[40]q +[5]q[8]q[4m+3]q3/bracketleftbigm+1 +3/bracketrightbig +q36[12]q3 +[3]q3[4]q3, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡27 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1 +5/bracketrightbig +q30[30]q +[5]q[6]q +×[3m+2]q4/bracketleftbig4m+3 +3/bracketrightbig +q9/bracketleftbigm+1 +2/bracketrightbig +q24[6]q4 +[2]q4[3]q4, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡28 (mod 30) ,then we have +Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1 +3/bracketrightbig +q18[6]q3 +[2]q3[3]q3 +×/bracketleftbig3m+2 +2/bracketrightbig +q8/bracketleftbig4m+3 +5/bracketrightbig +q15[15]q +[3]q[5]q[m+1]q12, +which, by Lemma 12, is a polynomial in qwith non-negative integer coefficients. If +m≡29 (mod 30) ,then we have +Catm(E6;q) =/bracketleftbig6m+1 +5/bracketrightbig +q10[10]q +[2]q[5]q[12m+5]q[2m+1]q6 +×[3m+2]q4[4m+3]q3/bracketleftbigm+1 +6/bracketrightbig +q72[72]q[3]q[4]q +[8]q[9]q[12]q, +which, by Corollary 6 and Lemma 7, is a polynomial in qwith non-negative integer +coefficients. +ForW=G36=E7, the degrees are 2 ,6,8,10,12,14,18, and hence +Catm(E7;q) =[18m+2]q[18m+6]q[18m+8]q[18m+10]q +[2]q[6]q[8]q[10]q +×[18m+12]q[18m+14]q[18m+18]q +[12]q[14]q[18]q.26 C. KRATTENTHALER AND T. W. M ¨ULLER +Ifm≡0 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +1 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +7/bracketrightbig +q14[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +2 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡3 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2 +×/bracketleftbig9m+5 +4/bracketrightbig +q8[3m+2]q6[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡4 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +20/bracketrightbig +q40[20]q2 +[4]q2[5]q2[9m+5]q2 +×/bracketleftbig3m+2 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡5 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +7/bracketrightbig +q14/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +6 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡7 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 27 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡8 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2/bracketleftbig9m+4 +4/bracketrightbig +q8 +×/bracketleftbig9m+5 +7/bracketrightbig +q14/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡9 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2/bracketleftbig9m+4 +5/bracketrightbig +q10 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡10 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14[3m+1]q6/bracketleftbig9m+4 +2/bracketrightbig +q4/bracketleftbig9m+5 +5/bracketrightbig +q10 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡11 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +4/bracketrightbig +q8 +×/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡12 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡13 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2,28 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡14 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +10/bracketrightbig +q20[10]q2 +[2]q2[5]q2 +×[9m+5]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18. +Ifonedecomposes[9 m+5]q2as[9m +2+3]q4+q2[9m +2+2]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡15 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×/bracketleftbig9m+5 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[3m+2]q6[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡16 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡17 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+4]q2/bracketleftbig9m+5 +2/bracketrightbig +q4 +×[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡18 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +28/bracketrightbig +q168[84]q2[2]q2 +[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18, +which, by Corollary 6 and Lemma 13, is a polynomial in qwith non-negative integer +coefficients. If m≡19 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[9m+5]q2 +×[3m+2]q6[9m+7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 29 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡20 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +2/bracketrightbig +q12 +×[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡21 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+4]q2/bracketleftbig9m+5 +2/bracketrightbig +q4 +×[3m+2]q6/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡22 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +2/bracketrightbig +q4/bracketleftbig9m+5 +7/bracketrightbig +q14/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4 +×/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡23 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +35/bracketrightbig +q210[105]q2 +[3]q2[5]q2[7]q2[9m+4]q2[9m+5]q2 +×[3m+2]q6[9m+7]q2/bracketleftbigm+1 +2/bracketrightbig +q36[6]q6 +[2]q6[3]q6, +which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer +coefficients. If m≡24 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14[3m+1]q6/bracketleftbig9m+4 +20/bracketrightbig +q40[20]q2 +[4]q2[5]q2[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡25 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +2/bracketrightbig +q12 +×[9m+4]q2/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18,30 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡26 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6/bracketleftbig9m+4 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡27 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡28 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡29 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10/bracketleftbig9m+5 +7/bracketrightbig +q14[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +30 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+4 +2/bracketrightbig +q4/bracketleftbig9m+5 +5/bracketrightbig +q10 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡31 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×/bracketleftbig9m+5 +4/bracketrightbig +q8[3m+2]q6[9m+7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 31 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡32 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8 +×[9m+5]q2/bracketleftbig3m+2 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡33 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2/bracketleftbig9m+4 +7/bracketrightbig +q14 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡34 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +10/bracketrightbig +q20[10]q2 +[2]q2[5]q2[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡35 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +36 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8/bracketleftbig9m+5 +7/bracketrightbig +q14/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +37 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +20/bracketrightbig +q40[20]q2 +[4]q2[5]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡38 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18,32 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡39 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10[9m+5]q2 +×/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡40 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2 +×/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡41 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡42 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+7 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡43 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2 +×/bracketleftbig9m+5 +7/bracketrightbig +q14[3m+2]q6[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡44 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+4 +20/bracketrightbig +q40[20]q2 +[4]q2[5]q2[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡45 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 33 +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +46 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +28/bracketrightbig +q168[84]q2[2]q2 +[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18, +which, by Lemma 13, is a polynomial in qwith non-negative integer coefficients. If +m≡47 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +7/bracketrightbig +q14[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +48 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡49 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+4 +5/bracketrightbig +q10/bracketleftbig9m+5 +2/bracketrightbig +q4 +×[3m+2]q6/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡50 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +2/bracketrightbig +q4 +×/bracketleftbig9m+5 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡51 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+4]q2 +×/bracketleftbig9m+5 +4/bracketrightbig +q8[3m+2]q6[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡52 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18,34 C. KRATTENTHALER AND T. W. M ¨ULLER +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +53 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2 +×[9m+5]q2/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡54 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +70/bracketrightbig +q140[70]q2 +[2]q2[5]q2[7]q2[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat, byCorollary6 +and Lemma 15, this is a polynomial in qwith non-negative integer coefficients. If +m≡55 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6 +×[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡56 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +57 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +2/bracketrightbig +q4/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4 +×[9m+4]q2/bracketleftbig9m+5 +7/bracketrightbig +q14[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡58 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +35/bracketrightbig +q210[105]q2 +[3]q2[5]q2[7]q2/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18, +which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer +coefficients. If m≡59 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10/bracketleftbig9m+5 +4/bracketrightbig +q8[3m+2]q6[9m+7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 35 +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +60 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8/bracketleftbig9m+5 +5/bracketrightbig +q10 +×/bracketleftbig3m+2 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡61 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +7/bracketrightbig +q14[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +62 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+7 +5/bracketrightbig +q10/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡63 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡64 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +20/bracketrightbig +q40[20]q2 +[4]q2[5]q2 +×/bracketleftbig9m+5 +7/bracketrightbig +q14/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡65 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+4]q2 +×/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡66 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[3m+1]q6/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18,36 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡67 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×[9m+5]q2/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡68 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2 +×/bracketleftbig9m+4 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2[9m+5]q2/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡69 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡70 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +2/bracketrightbig +q4/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4 +×/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡71 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +20/bracketrightbig +q40[20]q2 +[4]q2[5]q2/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×/bracketleftbig9m+5 +7/bracketrightbig +q14[3m+2]q6[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡72 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+4 +4/bracketrightbig +q8 +×[9m+5]q2/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 37 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡73 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡74 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +10/bracketrightbig +q20[10]q2 +[2]q2[5]q2 +×[9m+5]q2/bracketleftbig3m+2 +28/bracketrightbig +q168[84]q2[2]q2 +[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat, byCorollary6 +and Lemma 13, this is a polynomial in qwith non-negative integer coefficients. If +m≡75 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +7/bracketrightbig +q14/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6[9m+7]q2[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +76 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡77 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+4]q2 +×/bracketleftbig9m+5 +2/bracketrightbig +q4[3m+2]q6/bracketleftbig9m+7 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡78 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2 +×/bracketleftbig9m+4 +2/bracketrightbig +q4/bracketleftbig9m+5 +7/bracketrightbig +q14/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18,38 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡79 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2/bracketleftbig9m+4 +5/bracketrightbig +q10[9m+5]q2 +×[3m+2]q6[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡80 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +81 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×[9m+5]q2/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡82 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2 +×[9m+5]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18. +Ifonedecomposes[9 m+5]q2as[9m +2+4]q4+q2[9m +2+2]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡83 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2[9m+5]q +×[3m+2]q6[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡84 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +20/bracketrightbig +q40[20]q2 +[4]q2[5]q2 +×[9m+5]q2/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡85 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +2/bracketrightbig +q12 +×[9m+4]q2/bracketleftbig9m+5 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 39 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡86 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡87 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +4/bracketrightbig +q8[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +88 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡89 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡90 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +2/bracketrightbig +q4/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4 +×[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡91 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +4/bracketrightbig +q8[3m+2]q6/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +92 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8/bracketleftbig9m+5 +7/bracketrightbig +q14/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18,40 C. KRATTENTHALER AND T. W. M ¨ULLER +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +93 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +35/bracketrightbig +q210[105]q2 +[3]q2[5]q2[7]q2[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8/bracketleftbigm+1 +2/bracketrightbig +q36[6]q6 +[2]q6[3]q6, +which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer +coefficients. If m≡94 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14[3m+1]q6/bracketleftbig9m+4 +10/bracketrightbig +q20[10]q2 +[2]q2[5]q2[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡95 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +5/bracketrightbig +q10 +×/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡96 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6/bracketleftbig9m+4 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡97 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+4]q2/bracketleftbig9m+5 +2/bracketrightbig +q4 +×[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡98 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 41 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡99 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10/bracketleftbig9m+5 +7/bracketrightbig +q14[3m+2]q6[9m+7]q2[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +100 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+4 +4/bracketrightbig +q8 +×/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡101 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡102 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +28/bracketrightbig +q168[84]q2[2]q2 +[4]q2[6]q2[7]q2/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Lemma 13, is a polynomial in qwith non-negative integer coefficients. If +m≡103 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2/bracketleftbig9m+4 +7/bracketrightbig +q14[9m+5]q2 +×[3m+2]q6[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡104 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +20/bracketrightbig +q40[20]q2 +[4]q2[5]q2[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡105 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4 +×[9m+4]q2/bracketleftbig9m+5 +10/bracketrightbig +q20[10]q2 +[2]q2[5]q2[3m+2]q6/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18.42 C. KRATTENTHALER AND T. W. M ¨ULLER +If one decomposes [9 m+1]q2as [9m+1 +2]q4+q2[9m+1 +2]q4, then one sees that, by Corollary 6, +this is a polynomial in qwith non-negative integer coefficients. If m≡106 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6/bracketleftbig9m+4 +2/bracketrightbig +q4/bracketleftbig9m+5 +7/bracketrightbig +q14/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4 +×[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡107 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡108 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡109 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10 +×[9m+5]q2/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡110 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2 +×/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡111 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +4/bracketrightbig +q8[3m+2]q6 +×[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 43 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡112 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8 +×[9m+5]q2/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡113 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2 +×/bracketleftbig9m+5 +7/bracketrightbig +q14[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡114 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+4 +10/bracketrightbig +q20[10]q2 +[2]q2[5]q2[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡115 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2 +×/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6[9m+7]q[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡116 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡117 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+4 +7/bracketrightbig +q14/bracketleftbig9m+5 +2/bracketrightbig +q4 +×[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18,44 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡118 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡119 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +120 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8 +×/bracketleftbig9m+5 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡121 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡122 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14[3m+1]q6 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡123 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2[9m+5]q2 +×/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2[9m+7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 45 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡124 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2/bracketleftbigm+1 +5/bracketrightbig +q90[45]q2 +[5]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡125 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +2/bracketrightbig +q12 +×[9m+4]q2/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡126 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10[3m+1]q6 +×/bracketleftbig9m+4 +2/bracketrightbig +q4[9m+5]q2/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4/bracketleftbig9m+7 +7/bracketrightbig +q14[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡127 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12[9m+4]q2/bracketleftbig9m+5 +7/bracketrightbig +q14[3m+2]q6/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +128 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +35/bracketrightbig +q210[105]q2 +[3]q2[5]q2[7]q2/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Lemma 14, is a polynomial in qwith non-negative integer coefficients. If +m≡129 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +7/bracketrightbig +q14/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +4/bracketrightbig +q8[m+1]q18, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +130 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6 +×/bracketleftbig9m+4 +2/bracketrightbig +q4/bracketleftbig9m+5 +5/bracketrightbig +q10/bracketleftbig3m+2 +28/bracketrightbig +q168[84]q2[2]q2 +[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18, +which, by Lemma 13, is a polynomial in qwith non-negative integer coefficients. If +m≡131 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +5/bracketrightbig +q10/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +7/bracketrightbig +q14/bracketleftbig9m+5 +4/bracketrightbig +q8[3m+2]q6[9m+7]q2[m+1]q18,46 C. KRATTENTHALER AND T. W. M ¨ULLER +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +132 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8 +×[9m+5]q2/bracketleftbig3m+2 +2/bracketrightbig +q12/bracketleftbig9m+7 +5/bracketrightbig +q10/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡133 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +10/bracketrightbig +q60[30]q2 +[5]q2[6]q2[9m+4]q2 +×[9m+5]q2[3m+2]q6/bracketleftbig9m+7 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡134 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4 +10/bracketrightbig +q20[10]q2 +[2]q2[5]q2 +×/bracketleftbig9m+5 +7/bracketrightbig +q14/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡135 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[9m+4]q2 +×/bracketleftbig9m+5 +5/bracketrightbig +q10[3m+2]q6[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡136 (mod 140) ,then we have +Catm(E7;q) =/bracketleftbig9m+1 +35/bracketrightbig +q70[35]q2 +[5]q2[7]q2[3m+1]q6/bracketleftbig9m+4 +4/bracketrightbig +q8[9m+5]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q12[9m+7]q2[m+1]q18, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡137 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+4]q2/bracketleftbig9m+5 +2/bracketrightbig +q4 +×/bracketleftbig3m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig9m+7 +5/bracketrightbig +q10[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 47 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡138 (mod 140) ,then we have +Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1 +5/bracketrightbig +q30[15]q2 +[3]q2[5]q2/bracketleftbig9m+4 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2[9m+5]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q24[6]q4 +[2]q4[3]q4[9m+7]q2[m+1]q18. +Ifonedecomposes[9 m+7]q2as[9m +2+4]q4+q2[9m +2+3]q4, thenoneseesthat,byCorollary6, +this is a polynomial in qwith non-negative integer coefficients. If m≡139 (mod 140) , +then we have +Catm(E7;q) =/bracketleftbig9m+1 +4/bracketrightbig +q8/bracketleftbig3m+1 +2/bracketrightbig +q12/bracketleftbig9m+4 +5/bracketrightbig +q10[9m+5]q2 +×[3m+2]q6[9m+7]q2/bracketleftbigm+1 +7/bracketrightbig +q126[63]q2 +[7]q2[9]q2, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. +ForW=G37=E8, the degrees are 2 ,8,12,14,18,20,24,30, and hence +Catm(E7;q) =[30m+2]q[30m+8]q[30m+12]q[30m+14]q +[2]q[8]q[12]q[14]q +×[30m+18]q[30m+20]q[30m+24]q[30m+30]q +[18]q[20]q[24]q[30]q. +Ifm≡0 (mod 84),then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12/bracketleftbig15m+7 +7/bracketrightbig +q14 +×/bracketleftbig5m+3 +3/bracketrightbig +q18/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +1 (mod 84),then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2/bracketleftbig5m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡2 (mod 84),then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[m+1]q30,48 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡3 (mod 84),then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4/bracketleftbig15m+4 +7/bracketrightbig +q14[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡4 (mod 84),then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12 +×[15m+7]q2[5m+3]q6/bracketleftbig3m+2 +14/bracketrightbig +q140[70]q2 +[7]q2[10]q2/bracketleftbig5m+4 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡5 (mod 84),then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +28/bracketrightbig +q168[84]q2 +[7]q2[12]q2 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡6 (mod 84),then we have +Catm(E8;q) =/bracketleftbig15m+1 +7/bracketrightbig +q14/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2 +×/bracketleftbig5m+3 +3/bracketrightbig +q18/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡7 (mod 84),then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡8 (mod 84),then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +42/bracketrightbig +q252[126]q2[3]q2 +[6]q2[7]q2[9]q2[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 49 +which, by Lemma 22, is a polynomial in qwith non-negative integer coefficients. If +m≡9 (mod 84),then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6 +×[3m+2]q10/bracketleftbig5m+4 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡10 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2/bracketleftbig5m+2 +4/bracketrightbig +q24 +×[15m+7]q2[5m+3]q6/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[m+1]q30. +If one decomposes [15 m+ 7]q2as [15m +2+ 4]q4+q2[15m +2+ 3]q4, then one sees that, by +Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If m≡ +11 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×/bracketleftbig3m+2 +7/bracketrightbig +q70[35]q2 +[5]q2[7]q2[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 20, is a polynomial in qwith non-negative integer +coefficients. If m≡12 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12 +×[15m+7]q2/bracketleftbig5m+3 +21/bracketrightbig +q126[63]q2 +[7]q2[9]q2/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡13 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡14 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6/bracketleftbig15m+7 +7/bracketrightbig +q14 +×[5m+3]q6/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30,50 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡15 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2/bracketleftbig5m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡16 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +84/bracketrightbig +q504[252]q2[3]q2 +[7]q2[9]q2[12]q2[m+1]q30, +which, by Lemma 23, is a polynomial in qwith non-negative integer coefficients. If +m≡17 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8/bracketleftbig15m+4 +7/bracketrightbig +q14/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients. If +m≡18 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2/bracketleftbig5m+3 +3/bracketrightbig +q18 +/bracketleftbig3m+2 +28/bracketrightbig +q280[140]q2[2]q2 +[4]q2[7]q2[10]q2/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30, +which, by Lemma 24, is a polynomial in qwith non-negative integer coefficients. If +m≡19 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡20 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +7/bracketrightbig +q14/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 51 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡21 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2/bracketleftbig5m+3 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 17, is a polynomial in qwith non-negative integer +coefficients. If m≡22 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +28/bracketrightbig +q168[84]q2 +[7]q2[12]q2[15m+7]q2 +×[5m+3]q6/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡23 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10/bracketleftbig5m+4 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡24 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2/bracketleftbig5m+2 +2/bracketrightbig +q12[15m+7]q2 +×/bracketleftbig5m+3 +3/bracketrightbig +q18/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡25 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×/bracketleftbig3m+2 +7/bracketrightbig +q70[35]q2 +[5]q2[7]q2/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Lemma 18, is a polynomial in qwith non-negative integer coefficients. If +m≡26 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[15m+7]q2/bracketleftbig5m+3 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2 +×/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30. +If one decomposes [15 m+1]q2as [5m+1]q6+q2[5m]q6+q4[5m]q6, then one sees that, +by Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If52 C. KRATTENTHALER AND T. W. M ¨ULLER +m≡27 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 21, is a polynomial in qwith non-negative integer +coefficients. If m≡28 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12/bracketleftbig15m+7 +7/bracketrightbig +q14 +×[5m+3]q6/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡29 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2/bracketleftbig5m+2 +21/bracketrightbig +q126[63]q2 +[7]q2[9]q2/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡30 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2/bracketleftbig5m+3 +3/bracketrightbig +q18 +/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡31 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4/bracketleftbig15m+4 +7/bracketrightbig +q14[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Lemma 19, is a polynomial in qwith non-negative integer coefficients. If +m≡32 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +14/bracketrightbig +q140[70]q2 +[7]q2[10]q2/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 53 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡33 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +84/bracketrightbig +q504[252]q2[3]q2 +[7]q2[9]q2[12]q2 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Lemmas 16and23, is apolynomial in qwithnon-negative integer coefficients. +Ifm≡34 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +7/bracketrightbig +q14/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2 +[5m+3]q6/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡35 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡36 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[15m+7]q2/bracketleftbig5m+3 +3/bracketrightbig +q18 +×/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡37 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10/bracketleftbigg5m+4 +21/bracketrightbigg +q126[63]q2 +[7]q2[9]q2/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡38 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2/bracketleftbig5m+2 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6 +×[15m+7]q2[5m+3]q6/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30. +If one decomposes [15 m+ 7]q2as [15m +2+ 4]q4+q2[15m +2+ 3]q4, then one sees that, by +Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If m≡54 C. KRATTENTHALER AND T. W. M ¨ULLER +39 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6 +×/bracketleftbig3m+2 +7/bracketrightbig +q70[35]q2 +[5]q2[7]q2[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 20, is a polynomial in qwith non-negative integer +coefficients. If m≡40 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12[15m+7]q2/bracketleftbig5m+3 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2 +×/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[m+1]q30. +If one decomposes [15 m+7]q2as [5m+1]q6+q2[5m]q6+q4[5m]q6, then one sees that, +by Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If +m≡41 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡42 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24/bracketleftbig15m+7 +7/bracketrightbig +q14 +×/bracketleftbig5m+3 +3/bracketrightbig +q18/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡43 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2/bracketleftbig5m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡44 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +28/bracketrightbig +q168[84]q2 +[7]q2[12]q2[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 55 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡45 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8/bracketleftbig15m+4 +7/bracketrightbig +q14[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡46 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2 +[5m+3]q6/bracketleftbig3m+2 +28/bracketrightbig +q280[140]q2[2]q2 +[4]q2[7]q2[10]q2/bracketleftbig5m+4 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[m+1]q30, +which, by Corollary 6 and Lemma 24, is a polynomial in qwith non-negative integer +coefficients. If m≡47 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡48 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +7/bracketrightbig +q14/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12[15m+7]q2 +/bracketleftbig5m+3 +3/bracketrightbig +q18/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30, +which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡ +49 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡50 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +28/bracketrightbig +q168[84]q2 +[7]q2[12]q2[15m+7]q2 +×[5m+3]q6/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12/bracketleftbigm+1 +3/bracketrightbig +q90[15]q6 +[3]q6[5]q6,56 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡51 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6 +×[3m+2]q10/bracketleftbig5m+4 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡52 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2/bracketleftbig5m+2 +2/bracketrightbig +q12[15m+7]q2 +×[5m+3]q6/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡53 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×/bracketleftbig3m+2 +7/bracketrightbig +q70[35]q2 +[5]q2[7]q2[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Lemma 18, is a polynomial in qwith non-negative integer coefficients. If +m≡54 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2 +/bracketleftbig5m+3 +21/bracketrightbig +q126[63]q2 +[7]q2[9]q2/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡55 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Lemma 21, is a polynomial in qwith non-negative integer coefficients. If +m≡56 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6/bracketleftbig15m+7 +7/bracketrightbig +q14[5m+3]q6 +×/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 57 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡57 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2/bracketleftbig5m+2 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡58 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbigm+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2[5m+3]q6 +/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +42/bracketrightbig +q252[126]q2[3]q2 +[6]q2[7]q2[9]q2[m+1]q30, +which, by Lemma 22, is a polynomial in qwith non-negative integer coefficients. If +m≡59 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4/bracketleftbig15m+4 +7/bracketrightbig +q14/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Lemma 19, is a polynomial in qwith non-negative integer coefficients. If +m≡60 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12[15m+7]q2/bracketleftbig5m+3 +3/bracketrightbig +q18 +×/bracketleftbig3m+2 +14/bracketrightbig +q140[70]q2 +[7]q2[10]q2/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡61 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +28/bracketrightbig +q168[84]q2 +[7]q2[12]q2 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡62 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +7/bracketrightbig +q14/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30,58 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡63 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2/bracketleftbig5m+3 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡64 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +14/bracketrightbig +q84[42]q2 +[6]q2[7]q2[15m+7]q2 +×[5m+3]q6/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡65 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10/bracketleftbig5m+4 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡66 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2 +×/bracketleftbig5m+3 +3/bracketrightbig +q18/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡67 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×/bracketleftbig3m+2 +7/bracketrightbig +q70[35]q2 +[5]q2[7]q2/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Lemma 20, is a polynomial in qwith non-negative integer coefficients. If +m≡68 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6 +×[15m+7]q2/bracketleftbig5m+3 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 59 +If one decomposes [15 m+1]q2as [5m+1]q6+q2[5m]q6+q4[5m]q6, then one sees that, +by Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If +m≡69 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡70 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24/bracketleftbig15m+7 +7/bracketrightbig +q14 +×[5m+3]q6/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡71 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2/bracketleftbig5m+2 +21/bracketrightbig +q126[63]q2 +[7]q2[9]q2/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡72 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12[15m+7]q2/bracketleftbig5m+3 +3/bracketrightbig +q18/bracketleftbig3m+2 +2/bracketrightbig +q20 +/bracketleftbig5m+4 +28/bracketrightbig +q168[84]q2 +[7]q2[12]q2[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡73 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8/bracketleftbig15m+4 +7/bracketrightbig +q14[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10/bracketleftbig5m+4 +3/bracketrightbig +q18/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients. If +m≡74 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +28/bracketrightbig +q280[140]q2[2]q2 +[4]q2[7]q2[10]q2/bracketleftbig5m+4 +2/bracketrightbig +q12/bracketleftbigm+1 +3/bracketrightbig +q90[15]q6 +[3]q6[5]q6,60 C. KRATTENTHALER AND T. W. M ¨ULLER +which, by Corollary 6 and Lemma 24, is a polynomial in qwith non-negative integer +coefficients. If m≡75 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +42/bracketrightbig +q252[126]q2[3]q2 +[6]q2[7]q2[9]q2 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Lemmas 19and22, is apolynomial in qwithnon-negative integer coefficients. +Ifm≡76 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +7/bracketrightbig +q14/bracketleftbig15m+4 +4/bracketrightbig +q8/bracketleftbig5m+2 +2/bracketrightbig +q12[15m+7]q2 +×[5m+3]q6/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡77 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2/bracketleftbig5m+3 +4/bracketrightbig +q24 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer +coefficients. If m≡78 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +28/bracketrightbig +q168[84]q2 +[7]q2[12]q2[15m+7]q2/bracketleftbig5m+3 +3/bracketrightbig +q18 +×/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +2/bracketrightbig +q12[m+1]q30, +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡79 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +2/bracketrightbig +q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10/bracketleftbig5m+4 +21/bracketrightbig +q126[63]q2 +[7]q2[9]q2/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer +coefficients. If m≡80 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +28/bracketrightbig +q56[28]q2 +[4]q2[7]q2/bracketleftbig5m+2 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[15m+7]q2[5m+3]q6 +×/bracketleftbig3m+2 +2/bracketrightbig +q20/bracketleftbig5m+4 +4/bracketrightbig +q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 61 +which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If +m≡81 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +4/bracketrightbig +q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7 +2/bracketrightbig +q4/bracketleftbig5m+3 +12/bracketrightbig +q72[12]q6 +[3]q6[4]q6 +×/bracketleftbig3m+2 +7/bracketrightbig +q70[35]q2 +[5]q2[7]q2[5m+4]q6/bracketleftbigm+1 +2/bracketrightbig +q60[30]q2[2]q2[3]q2[5]q2 +[6]q2[10]q2[15]q2, +which, by Corollary 6 and Lemma 18, is a polynomial in qwith non-negative integer +coefficients. If m≡82 (mod 84) ,then we have +Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4 +2/bracketrightbig +q4/bracketleftbig5m+2 +4/bracketrightbig +q24[15m+7]q2/bracketleftbig5m+3 +7/bracketrightbig +q42[21]q2 +[3]q2[7]q2 +/bracketleftbig3m+2 +4/bracketrightbig +q40[10]q4 +[2]q4[5]q4/bracketleftbig5m+4 +6/bracketrightbig +q36[6]q6 +[2]q6[3]q6[m+1]q30. +If one decomposes [15 m+1]q2as [5m+1]q6+q2[5m]q6+q4[5m]q6, then one sees that, +by Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If +m≡83 (mod 84) ,then we have +Catm(E8;q) =/bracketleftbig15m+1 +14/bracketrightbig +q28[14]q2 +[2]q2[7]q2[15m+4]q2/bracketleftbig5m+2 +3/bracketrightbig +q18/bracketleftbig15m+7 +4/bracketrightbig +q8/bracketleftbig5m+3 +2/bracketrightbig +q12 +×[3m+2]q10[5m+4]q6/bracketleftbigm+1 +4/bracketrightbig +q120[60]q2[2]q2[3]q2[5]q2 +[10]q2[12]q2[15]q2, +which, by Corollary 6 and Lemma 21, is a polynomial in qwith non-negative integer +coefficients. +/square +5.Auxiliary results I +This section collects several auxiliary results which allow us to reduce the problem +of proving Theorem 2, or the equivalent statement (3.3), for the 2 6 exceptional groups +listed in Section 2 to a finite problem. While Lemmas 27 and 28 cover spec ial choices +of the parameters, Lemmas 26 and 30 afford an inductive procedur e. More precisely, +if we assume that we have already verified Theorem 2 for all groups o f smaller rank, +then Lemmas 26 and 30, together with Lemmas 27 and 31, reduce th e verification of +Theorem 2 for the group that we are currently considering to a finit e problem; see +Remark 3. The final lemma of this section, Lemma 32, disposes of com plex reflection +groups with a special property satisfied by their degrees. +Letp=am+b, 0≤bnthen +FixNCm(W)(φp) =/braceleftbig +(c;ε,...,ε)/bracerightbig +. +Proof.Let us suppose that ( w0;w1,...,w m)∈FixNCm(W)(φp) and that there exists a +j≥1 such that wj/ne}ationslash=ε. By (5.8), it then follows for such a jthat alsowk/ne}ationslash=εfor +allk≡j−lm1b(modm), where, as before, bis defined as the unique integer with +h1=am2+band 0≤b < m 2. Since, by assumption, gcd( b,m2) = 1, there are +exactlym2suchk’s which are distinct mod m. However, this implies that the sum of +the absolute lengths of the wi’s, 0≤i≤m, is at least m2> n, a contradiction to +Remark 1.(2). /square +Remark 3.(1) If we put ourselves in the situation of the assumptions of Lemma 30, +then we may conclude that equation (3.3) only needs to be checked f or pairs (m2,h2) +subject to the following restrictions: +m2≥2,gcd(h1,m2) = 1,andh2divides all degrees of W. (5.9) +Indeed, Lemmas 27 and 30 together imply that equation (3.3) is alway s satisfied in all +other cases. +(2) Still putting ourselves in the situation of Lemma 30, if m2>nandm2h2does not +divide any of the degrees of W, then equation (3.3) is satisfied. Indeed, Lemma 31 says +thatinthiscasetheleft-handsideof (3.3)equals1,whileastraightf orwardcomputation +using (5.4) shows that in this case the right-hand side of (3.3) equals 1 as well. +(3)It shouldbeobserved that thisleaves afinitenumber of choices form2to consider, +whence a finite number of choices for ( m1,m2,h1,h2). Altogether, there remains a finite +number of choices for p=h1m1to be checked. +Lemma 32. LetWbe an irreducible well-generated complex reflection group o f rankn +with the property that di|hfori= 1,2,...,n. Then Theorem 2is true for this group +W. +Proof.By Lemma 26, we may restrict ourselves to divisors pofmh.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 65 +Suppose that e2πip/mhis adi-th rootof unity for some i. In other words, mh/pdivides +di. Sincediis a divisor of hby assumption, the integer mh/palso divides h. But this +is equivalent to saying that mdividesp, and equation (3.3) holds by Lemma 27. +Now assume that mh/pdoes not divide any of the di’s. Then, by (5.4), the right- +hand side of (3.3) equals 1. On the other hand, ( c;ε,...,ε) is always an element of +FixNCm(W)(φp). To see that there are no others, we make appeal to the classific a- +tion of all irreducible well-generated complex reflection groups, whic h we recalled in +Section 2. Inspection reveals that all groups satisfying the hypot heses of the lemma +have rank n≤2. Except for the groups contained in the infinite series G(d,1,n) +andG(e,e,n) for which Theorem 2 has been established in [19], these are the grou ps +G5,G6,G9,G10,G14,G17,G18,G21. We now discuss these groups case by case, keeping +the notation of Lemma 30. In order to simplify the argument, we not e that Lemma 31 +implies that equation (3.3) holds if m2>2, so that in the following arguments we +always may assume that m2= 2. +CaseG5. The degrees are 6 ,12, and therefore Remark 3.(1) implies that equa- +tion (3.3) is always satisfied. +CaseG6. The degrees are 4 ,12, and therefore, according to Remark 3.(1), we need +only consider the casewhere h2= 4andm2= 2, that is, p= 3m/2. Then (5.8) becomes +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c2wm +2+1c−2,c2wm +2+2c−2,...,c2wmc−2,cw1c−1,...,cw m +2c−1/parenrightbig +. +(5.10) +If (w0;w1,...,w m) isfixed by φpandnot equal to ( c;ε,...,ε), there must exist an iwith +1≤i≤m +2such thatℓT(wi) =ℓT(wm +2+i) = 1,wm +2+i=cwic−1,wiwm +2+i=wicwic−1=c, +and allwj, withj/ne}ationslash=i,m +2+i, equalε. However, with the help of the GAPpackage +CHEVIE[14, 28], one verifies that there is no wiinG6such that +ℓT(wi) = 1 and wicwic−1=c +are simultaneously satisfied. Hence, the left-hand side of (3.3) is eq ual to 1, as required. +CaseG9. The degrees are 8 ,24, and therefore, according to Remark 3.(1), we need +only consider the case where h2= 8 andm2= 2, that is, p= 3m/2. This is the same p +as forG6. Again, CHEVIEfinds no solution. Hence, the left-hand side of (3.3) is equal +to 1, as required. +CaseG10. The degrees are 12 ,24, and therefore Remark 3.(1) implies that equa- +tion (3.3) is always satisfied. +CaseG14. The degrees are 6 ,24, and therefore Remark 3.(1) implies that equa- +tion (3.3) is always satisfied. +CaseG17. The degrees are 20 ,60, and therefore, according to Remark 3.(1), we need +only consider the cases where h2= 20 orh2= 4. In the first case, p= 3m/2, which is +the samepas forG6. Again,CHEVIEfinds no solution. In the second case, p= 15m/2. +Then (5.8) becomes +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c8wm +2+1c−8,c8wm +2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm +2c−7/parenrightbig +.(5.11) +By Lemma 29, every element of NC(W) is fixed under conjugation by c3, and, thus, on +elements fixed by φp, the above action of φpreduces to the one in (5.10). This action66 C. KRATTENTHALER AND T. W. M ¨ULLER +was already discussed in the first case. Hence, in both cases, the le ft-hand side of (3.3) +is equal to 1, as required. +CaseG18. The degrees are 30 ,60, and therefore Remark 3.(1) implies that equa- +tion (3.3) is always satisfied. +CaseG21. The degrees are 12 ,60, and therefore, according to Remark 3.(1), we need +only consider the cases where h2= 12 orh2= 4. In the first case, p= 5m/2, so that +(5.8) becomes +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c3wm +2+1c−3,c3wm +2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm +2c−2/parenrightbig +.(5.12) +If (w0;w1,...,w m) is fixed by φpand not equal to ( c;ε,...,ε), there must exist an i +with 1≤i≤m +2such thatℓT(wi) = 1 andwic2wic−2=c. However, with the help of +theGAPpackageCHEVIE[14, 28], one verifies that there is no such solution to this +equation. In the second case, p= 15m/2. Then (5.8) becomes the action in (5.11). +By Lemma 29, every element of NC(W) is fixed under conjugation by c5, and, thus, +on elements fixed by φp, the action of φpin (5.11) reduces to the one in the first case. +Hence, in both cases, the left-hand side of (3.3) is equal to 1, as re quired. +This completes the proof of the lemma. /square +6.Case-by-case verification of Theorem 2 +In the sequel we write ζdfor a primitive d-th root of unity. +CaseG4.The degrees are 4 ,6, and hence we have +Catm(G4;q) =[6m+6]q[6m+4]q +[6]q[4]q. +Letζbe a 6m-th root of unity. In what follows, we abbreviate the assertion tha t “ζis +a primitive d-th root of unity” as “ ζ=ζd.” The following cases on the right-hand side +of (3.3) occur: +lim +q→ζCatm(G4;q) =m+1,ifζ=ζ6,ζ3, (6.1a) +lim +q→ζCatm(G4;q) =3m+2 +2,ifζ=ζ4,2|m, (6.1b) +lim +q→ζCatm(G4;q) = Catm(G4),ifζ=−1 orζ= 1, (6.1c) +lim +q→ζCatm(G4;q) = 1,otherwise. (6.1d) +We must now prove that the left-hand side of (3.3) in each case agre es with the +values exhibited in (6.1). The only cases not covered by Lemmas 27 an d 28 are the +ones in (6.1b) and (6.1d). On the other hand, the only case left to co nsider according +to Remark 3 is the case where h2=m2= 2, that is the case (6.1b) where p= 3m/2. In +particular, mmust be divisible by 2. The action of φpis the same as the one in (5.10). +With the help of CHEVIE, one finds that each of the 3 (complex) reflections in G4which +are less than the (chosen) Coxeter element is a valid choice for wi, and each of these +choices gives rise to m/2 elements in NCm(G4) since the index iranges from 1 to m/2. +Hence, in total, we obtain 1+3m +2=3m+2 +2elements in Fix NCm(G4)(φp), which agrees +with the limit in (6.1b).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 67 +CaseG8.The degrees are 8 ,12, and hence we have +Catm(G8;q) =[12m+12]q[12m+8]q +[12]q[8]q. +Letζbe a 12m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(G8;q) =m+1,ifζ=ζ12,ζ6,ζ3, (6.2a) +lim +q→ζCatm(G8;q) =3m+2 +2,ifζ=ζ8,2|m, (6.2b) +lim +q→ζCatm(G8;q) = Catm(G8),ifζ=ζ4,−1,1, (6.2c) +lim +q→ζCatm(G8;q) = 1,otherwise. (6.2d) +We must now prove that the left-hand side of (3.3) in each case agre es with the +values exhibited in (6.2). The only cases not covered by Lemmas 27 an d 28 are the +ones in (6.2b) and (6.2d). On the other hand, the only case left to co nsider according to +Remark 3 is the case where h2= 4 andm2= 2, that is the case (6.2b) where p= 3m/2. +In particular, mmust be divisible by 2. The action of φpis the same as the one in +(5.10). With the help of CHEVIE, one finds that each of the 3 (complex) reflections in +G8which are less than the (chosen) Coxeter element is a valid choice for wi, and each +of these choices gives rise to m/2 elements in NCm(G8) since the index iranges from +1 tom/2. +Hence, in total, we obtain 1+3m +2=3m+2 +2elements in Fix NCm(G8)(φp), which agrees +with the limit in (6.2b). +CaseG16.The degrees are 20 ,30, and hence we have +Catm(G16;q) =[30m+30]q[30m+20]q +[30]q[20]q. +Letζbe a 30m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(G16;q) =m+1,ifζ=ζ30,ζ15,ζ6,ζ3, (6.3a) +lim +q→ζCatm(G16;q) =3m+2 +2,ifζ=ζ20,ζ4,2|m, (6.3b) +lim +q→ζCatm(G16;q) = Catm(G16),ifζ=ζ10,ζ5,−1,1, (6.3c) +lim +q→ζCatm(G16;q) = 1,otherwise. (6.3d) +We must now prove that the left-hand side of (3.3) in each case agre es with the +values exhibited in (6.3). The only cases not covered by Lemmas 27 an d 28 are the +ones in (6.3b) and (6.3d). On the other hand, the only cases left to c onsider according +to Remark 3 are the cases where h2= 10 andm2= 2, respectively h2=m2= 2. Both +cases belong to (6.3b). In the first case, we have p= 3m/2, while in the second case we +havep= 15m/2. In particular, mmust be divisible by 2. In the first case, the action +ofφpis the same as the one in (5.10). With the help of CHEVIE, one finds that each of +the 3 (complex) reflections in G16which are less than the (chosen) Coxeter element is +a valid choice for wi, and each of these choices gives rise to m/2 elements in NCm(G16)68 C. KRATTENTHALER AND T. W. M ¨ULLER +since the index iranges from 1 to m/2. On the other hand, if p= 15m/2, then the +action ofφpis the same as the one in (5.11). By Lemma 29, every element of NC(W) +is fixed under conjugation by c3, and, thus, on elements fixed by φp, the action of φp +reduces to the one in the first case. +Hence, in total, we obtain 1+3m +2=3m+2 +2elements in Fix NCm(G16)(φp), which agrees +with the limit in (6.3b). +CaseG20.The degrees are 12 ,30, and hence we have +Catm(G20;q) =[30m+30]q[30m+12]q +[30]q[12]q. +Letζbe a 30m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(G20;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (6.4a) +lim +q→ζCatm(G20;q) =5m+2 +2,ifζ=ζ12,ζ4,2|m, (6.4b) +lim +q→ζCatm(G20;q) = Catm(G20),ifζ=ζ6,ζ3,−1,1, (6.4c) +lim +q→ζCatm(G20;q) = 1,otherwise. (6.4d) +We must now prove that the left-hand side of (3.3) in each case agre es with the +values exhibited in (6.4). The only cases not covered by Lemmas 27 an d 28 are the +ones in (6.4b) and (6.4d). On the other hand, the only cases left to c onsider according +to Remark 3 are the cases where h2= 6 andm2= 2, respectively h2=m2= 2. Both +cases belong to (6.4b). In the first case, we have p= 5m/2, while in the second case we +havep= 15m/2. In particular, mmust be divisible by 2. In the first case, the action +ofφpis the same as the one in (5.12). With the help of CHEVIE, one finds that each of +the 5 (complex) reflections in G20which are less than the (chosen) Coxeter element is +a valid choice for wi, and each of these choices gives rise to m/2 elements in NCm(G20) +since the index iranges from 1 to m/2. On the other hand, if p= 15m/2, then the +action ofφpis the same as the one in (5.11). By Lemma 29, every element of NC(W) +is fixed under conjugation by c5, and, thus, on elements fixed by φp, the action of φp +reduces to the one in the first case. +Hence, in total, we obtain 1+5m +2=5m+2 +2elements in Fix NCm(G20)(φp), which agrees +with the limit in (6.4b). +CaseG23=H3.The degrees are 2 ,6,10, and hence we have +Catm(H3;q) =[10m+10]q[10m+6]q[10m+2]q +[10]q[6]q[2]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 69 +Letζbe a 10m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(H3;q) =m+1,ifζ=ζ10,ζ5, (6.5a) +lim +q→ζCatm(H3;q) =10m+6 +6,ifζ=ζ6,ζ3,3|m, (6.5b) +lim +q→ζCatm(H3;q) = Catm(H3),ifζ=−1 orζ= 1, (6.5c) +lim +q→ζCatm(H3;q) = 1,otherwise. (6.5d) +We must now prove that the left-handside of (3.3) in each case agre es with the values +exhibited in (6.5). The only cases not covered by Lemmas 27 and 28 ar e the ones in +(6.5b) and (6.5d). By Lemma 26, we are free to choose p= 5m/3 ifζ=ζ6, respectively +p= 10m/3 ifζ=ζ3. In both cases, mmust be divisible by 3. +We start with the case that p= 5m/3. From (5.1), we infer +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c2wm +3+1c−2,c2wm +3+2c−2,...,c2wmc−2,cw1c−1,...,cw m +3c−1/parenrightbig +. +(6.6) +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c2wm +3+ic−2, i= 1,2,...,2m +3, (6.7a) +wi=cwi−2m +3c−1, i=2m +3+1,2m +3+2,...,m. (6.7b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +3such that +ℓT(wi) =ℓT(wi+m +3) =ℓT(wi+2m +3) = 1. +Writingt1,t2,t3forwi,wi+m +3,wi+2m +3, respectively, the equations (6.7) reduce to +t1=c2t2c−2, (6.8a) +t2=c2t3c−2, (6.8b) +t3=ct1c−1. (6.8c) +One of these equations is in fact superfluous: if we substitute (6.8b ) and (6.8c) in +(6.8a), then we obtain t1=c5t1c−5which is automatically satisfied due to Lemma 29 +withd= 2. +Since (w0;w1,...,w m)∈NCm(H3), we must have t1t2t3=c. Combining this with +(6.8), we infer that +t1(c−2t1c2)(ct1c−1) =c. (6.9) +With the help of Stembridge’s Maplepackagecoxeter [36], one obtains five solutions +fort1in this equation: +t1∈/braceleftbig +[2],[3],[2,1,2],[1,2,3,2,1],[1,3,2,1,2,1,3]/bracerightbig +. (6.10) +Here we have used the short notation of coxeter: if{s1,s2,s3}is a simple system of +generators of H3, corresponding to the Dynkin diagram displayed in Figure 1, then +[j1,j2,...,j k] stands for the element sj1sj2...sjk. +We claim that each of the above five solutions gives rise to m/3 elements of +FixNCm(H3)(φp). Indeed, given t1, the elements t2andt3can be computed by (6.8a) +and (6.8c), and there are m/3 possibilities to choose the index iforwi.70 C. KRATTENTHALER AND T. W. M ¨ULLER +• • •1 2 35 +Figure 1. The Dynkin diagram for H3 +In total, we obtain 1 + 5m +3=10m+6 +6elements in Fix NCm(H3)(φp), which agrees with +the limit in (6.5b). +In the case that p= 10m/3, we infer from (5.1) that +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c4w2m +3+1c−4,c4w2m +3+2c−4,...,c4wmc−4,c3w1c−3,...,c3w2m +3c−3/parenrightbig +.(6.11) +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c4w2m +3+ic−4, i= 1,2,...,m +3, (6.12a) +wi=c3wi−m +3c−3, i=m +3+1,m +3+2,...,m. (6.12b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +3such that +ℓT(wi) =ℓT(wi+m +3) =ℓT(wi+2m +3) = 1. +Writingt1,t2,t3forwi,wi+m +3,wi+2m +3, respectively, the equations (6.12) reduce to +t1=c4t3c−4, (6.13a) +t2=c3t1c−3, (6.13b) +t3=c3t2c−3. (6.13c) +One of these equations is in fact superfluous: if we substitute (6.13 b) and (6.13c) in +(6.13a), then we obtain t1=c10t1c−10which is automatically satisfied since c10=ε. +Since (w0;w1,...,w m)∈NCm(H3), we must have t1t2t3=c. Combining this with +(6.13), we infer that +t1(c3t1c−3)(c−4t1c4) =c. (6.14) +Using that c5t1c−5=t1, due to Lemma 29 with d= 2, we see that this equation is +equivalent with (6.9). Therefore, we are facing exactly the same en umeration problem +here as forp= 5m/3, and, consequently, the number of solutions to (6.14) is the same , +namely5m+3 +3, as required. +Finally, we turn to (6.5d). By Remark 3, the only choices for h2andm2to be +considered are h2= 1 andm2= 3,h2=m2= 2, respectively h2= 2 andm2= 3. +These correspond to the choices p= 10m/3,p= 5m/2, respectively p= 5m/3, out of +which only p= 5m/2 has not yet been discussed and belongs to the current case. The +corresponding action of φpis given by (5.12). A computation with Stembridge’s Maple +packagecoxeter [36] finds no solution. Hence, the left-hand side of (3.3) is equal to 1 , +as required.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 71 +CaseG24.The degrees are 4 ,6,14, and hence we have +Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q +[14]q[6]q[4]q. +Letζbe a 14m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (6.15a) +lim +q→ζCatm(G24;q) =7m+3 +3,ifζ=ζ6,ζ3,3|m, (6.15b) +lim +q→ζCatm(G24;q) =7m+2 +2,ifζ=ζ4,2|m, (6.15c) +lim +q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (6.15d) +lim +q→ζCatm(G24;q) = 1,otherwise. (6.15e) +We must now prove that the left-handside of (3.3) in each case agre es with the values +exhibited in (6.15). The only cases not covered by Lemmas 27 and 28 a re the ones in +(6.15b), (6.15c), and (6.15e). +We first consider (6.15b). By Lemma 26, we are free to choose p= 7m/3 ifζ=ζ6, +respectively p= 14m/3 ifζ=ζ3. In both cases, mmust be divisible by 3. +We start with the case that p= 7m/3. From (5.1), we infer +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c3w2m +3+1c−3,c3w2m +3+2c−3,...,c3wmc−3,c2w1c−2,...,c2w2m +3c−2/parenrightbig +. +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c3w2m +3+ic−3, i= 1,2,...,m +3, (6.16a) +wi=c2wi−m +3c−2, i=m +3+1,m +3+2,...,m. (6.16b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +3such that +ℓT(wi) =ℓT(wi+m +3) =ℓT(wi+2m +3) = 1. +Writingt1,t2,t3forwi,wi+m +3,wi+2m +3, respectively, the equations (6.16) reduce to +t1=c3t3c−3, (6.17a) +t2=c2t1c−2, (6.17b) +t3=c2t2c−2. (6.17c) +One of these equations is in fact superfluous: if we substitute (6.17 b) and (6.17c) in +(6.17a), then we obtain t1=c7t1c−7which is automatically satisfied due to Lemma 29 +withd= 2. +Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2t3=c. Combining this with +(6.17), we infer that +t1(c2t1c−2)(c4t1c−4) =c. (6.18) +With the help of CHEVIE, one obtains 7 solutions for t1in this equation: +t1∈/braceleftbig +[1],[2],[3],[15],[16],[19],[21]/bracerightbig +, (6.19)72 C. KRATTENTHALER AND T. W. M ¨ULLER +each of them giving rise to m/3 elements of Fix NCm(G24)(φp) sinceiranges from 1 to +m/3. Here we have used the short notation of CHEVIE: [j1,j2,...,j k] stands for the +elementrj1rj2...rjk, whereriis thei-th (complex) reflection corresponding to the i-th +root in the internal ordering of the roots of G24inCHEVIE. +In total, we obtain 1 + 7m +3=7m+3 +3elements in Fix NCm(G24)(φp), which agrees with +the limit in (6.15b). +In the case that p= 14m/3, we infer from (5.1) that +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c5wm +3+1c−5,c5wm +3+2c−5,...,c5wmc−5,c4w1c−4,...,c4wm +3c−4/parenrightbig +. +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c5wm +3+ic−5, i= 1,2,...,2m +3, (6.20a) +wi=c4wi−2m +3c−4, i=2m +3+1,2m +3+2,...,m. (6.20b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +3such that +ℓT(wi) =ℓT(wi+m +3) =ℓT(wi+2m +3) = 1. +Writingt1,t2,t3forwi,wi+m +3,wi+2m +3, respectively, the equations (6.20) reduce to +t1=c5t2c−5, (6.21a) +t2=c5t3c−5, (6.21b) +t3=c4t1c−4. (6.21c) +One of these equations is in fact superfluous: if we substitute (6.21 b) and (6.21c) in +(6.21a), then we obtain t1=c14t1c−14which is automatically satisfied since c14=ε. +Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2t3=c. Combining this with +(6.21), we infer that +t1(c9t1c−9)(c−4t1c4) =c. (6.22) +Using that c7t1c−7=t1, due to Lemma 29 with d= 2, we see that this equation is +equivalent with (6.18). Therefore, we are facing exactly the same e numeration problem +here as forp= 7m/3, and, consequently, the number of solutions to (6.22) is the same , +namely7m+3 +3, as required. +Ournextcaseis(6.15c). ByLemma26, wearefreetochoose p= 7m/2. Inparticular, +mmust be divisible by 2. From (5.1), we infer +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c4wm +2+1c−4,c4wm +2+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm +2c−3/parenrightbig +. +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c4wm +2+ic−4, i= 1,2,...,m +2, (6.23a) +wi=c3wi−m +2c−3, i=m +2+1,m +2+2,...,m. (6.23b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +2such that +ℓT(wi) =ℓT(wi+m +2) = 1.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 73 +Writingt1,t2forwi,wi+m +2, respectively, the equations (6.23) reduce to +t1=c4t2c−4, (6.24a) +t2=c3t1c−3. (6.24b) +One of these equations is in fact superfluous: if we substitute (6.24 b) in (6.24a), then +we obtaint1=c7t1c−7which is automatically satisfied due to Lemma 29 with d= 2. +Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2≤Tc, where≤Tis the partial +order defined in (2.1). Combining this with (6.24), we infer that +t1(c3t1c−3)≤Tc. (6.25) +With the help of CHEVIE, one obtains 7 solutions for t1in this relation: +t1∈/braceleftbig +[5],[6],[7],[9],[12],[29],[32]/bracerightbig +, (6.26) +each of them giving rise to m/2 elements of Fix NCm(G24)(φp) sinceiranges from 1 to +m/2. Here we have used again the short notation of CHEVIEreferring to the internal +ordering of the roots of G24inCHEVIE. +In total, we obtain 1 + 7m +2=7m+2 +2elements in Fix NCm(G24)(φp), which agrees with +the limit in (6.15c). +Finally, we turn to (6.15e). By Remark 3, the only choices for h2andm2to be con- +sidered are h2= 1 andm2= 3,h2=m2= 2, andh2= 2 andm2= 3. These correspond +to the choices p= 14m/3,p= 7m/2, respectively p= 7m/3, all of which have already +been discussed as they do not belong to (6.15e). Hence, (3.3) must necessarily hold, as +required. +CaseG25.The degrees are 6 ,9,12, and hence we have +Catm(G25;q) =[12m+12]q[12m+9]q[12m+6]q +[12]q[9]q[6]q. +Letζbe a 12m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(G25;q) =m+1,ifζ=ζ12,ζ4, (6.27a) +lim +q→ζCatm(G25;q) =4m+3 +3,ifζ=ζ9,3|m, (6.27b) +lim +q→ζCatm(G25;q) = (m+1)(2m+1),ifζ=ζ6,−1 (6.27c) +lim +q→ζCatm(G25;q) = Catm(G25),ifζ=ζ3,1, (6.27d) +lim +q→ζCatm(G25;q) = 1,otherwise. (6.27e) +We must now prove that the left-handside of (3.3) in each case agre es with the values +exhibited in (6.27). The only cases not covered by Lemmas 27 and 28 a re the ones in +(6.27b) and (6.27e). +We first consider (6.27b). By Lemma 26, we are free to choose p= 4m/3. In +particular, mmust be divisible by 3. From (5.1), we infer +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c2w2m +3+1c−2,c2w2m +3+2c−2,...,c2wmc−2,cw1c−1,...,cw 2m +3c−1/parenrightbig +.74 C. KRATTENTHALER AND T. W. M ¨ULLER +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c2w2m +3+ic−2, i= 1,2,...,m +3, (6.28a) +wi=cwi−m +3c−1, i=m +3+1,m +3+2,...,m. (6.28b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +3such that +ℓT(wi) =ℓT(wi+m +3) =ℓT(wi+2m +3) = 1. +Writingt1,t2,t3forwi,wi+m +3,wi+2m +3, respectively, the equations (6.28) reduce to +t1=c2t3c−2, (6.29a) +t2=ct1c−1, (6.29b) +t3=ct2c−1. (6.29c) +One of these equations is in fact superfluous: if we substitute (6.29 b) and (6.29c) in +(6.29a), then we obtain t1=c4t1c−4which is automatically satisfied due to Lemma 29 +withd= 3. +Since (w0;w1,...,w m)∈NCm(G25), we must have t1t2t3=c. Combining this with +(6.29), we infer that +t1(ct1c−1)(c2t1c−2) =c. (6.30) +With the help of CHEVIE, one obtains four solutions for t1in this equation: +t1∈/braceleftbig +[1],[2],[3],[14]/bracerightbig +, (6.31) +each of them giving rise to m/3 elements of Fix NCm(G25)(φp) sinceiranges from 1 to +m/3. Here we have used again the short notation of CHEVIEreferring to the internal +ordering of the roots of G25inCHEVIE. +In total, we obtain 1 + 4m +3=4m+3 +3elements in Fix NCm(G25)(φp), which agrees with +the limit in (6.27b). +Finally, we turn to (6.27e). By Remark 3, the only choice for h2andm2to be +considered are h2=m2= 3. This corresponds to the choice p= 4m/3, which has +already been discussed as they do not belong to (6.27e). Hence, (3 .3) must necessarily +hold, as required. +CaseG26.The degrees are 6 ,12,18, and hence we have +Catm(G26;q) =[18m+18]q[18m+12]q[18m+6]q +[18]q[12]q[6]q. +Letζbe a 14m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(G26;q) =m+1,ifζ=ζ18,ζ9, (6.32a) +lim +q→ζCatm(G26;q) =3m+2 +2,ifζ=ζ12,ζ4,2|m, (6.32b) +lim +q→ζCatm(G26;q) = Catm(G26),ifζ=ζ6,ζ3,−1,1, (6.32c) +lim +q→ζCatm(G26;q) = 1,otherwise. (6.32d)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 75 +We must now prove that the left-handside of (3.3) in each case agre es with the values +exhibited in (6.32). The only cases not covered by Lemmas 27 and 28 a re the ones in +(6.32b) and (6.32d). +We first consider (6.32b). By Lemma 26, we are free to choose p= 3m/2 ifζ=ζ12, +respectively p= 9m/2 ifζ=ζ4. In both cases, mmust be divisible by 2. +We start with the case that p= 3m/2. From (5.1), we infer +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c2wm +2+1c−2,c2wm +2+2c−2,...,c2wmc−2,cw1c−1,...,cw m +2c−1/parenrightbig +. +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c2wm +2+ic−2, i= 1,2,...,m +2, (6.33a) +wi=cwi−m +2c−1, i=m +2+1,m +2+2,...,m. (6.33b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +2such that +ℓT(wi) =ℓT(wi+m +2) = 1. +Writingt1,t2forwi,wi+m +2, respectively, the equations (6.33) reduce to +t1=c2t2c−2, (6.34a) +t2=c1t1c−1. (6.34b) +One of these equations is in fact superfluous: if we substitute (6.34 b) in (6.34a), then +we obtaint1=c3t1c−3which is automatically satisfied due to Lemma 29 with d= 6. +Since (w0;w1,...,w m)∈NCm(G26), we must have t1t2≤Tc. Combining this with +(6.34), we infer that +t1(ct1c−1)≤Tc. (6.35) +With the help of CHEVIE, one obtains three solutions for t1in this equation: +t1∈/braceleftbig +[2],[3],[12]/bracerightbig +, +each of them giving rise to m/2 elements of Fix NCm(G26)(φp) sinceiranges from 1 to +m/2. Here we have again used the short notation of CHEVIEreferring to the internal +ordering of the roots of G26inCHEVIE. +In total, we obtain 1 + 3m +2=3m+2 +2elements in Fix NCm(G26)(φp), which agrees with +the limit in (6.32b). +In the case that p= 9m/2, we infer from (5.1) that +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c5wm +2+1c−5,c5wm +2+2c−5,...,c5wmc−5,c4w1c−4,...,c4wm +2c−4/parenrightbig +. +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c5wm +2+ic−5, i= 1,2,...,m +2, (6.36a) +wi=c4wi−m +2c−4, i=m +2+1,m +2+2,...,m. (6.36b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +3such that +ℓT(wi) =ℓT(wi+m +2) = 1.76 C. KRATTENTHALER AND T. W. M ¨ULLER +Writingt1,t2forwi,wi+m +2, respectively, the equations (6.36) reduce to +t1=c5t2c−5, (6.37a) +t2=c4t1c−4. (6.37b) +One of these equations is in fact superfluous: if we substitute (6.37 b) in (6.37a), then +we obtaint1=c9t1c−9which is automatically satisfied due to Lemma 29 with d= 2. +Since (w0;w1,...,w m)∈NCm(G26), we must have t1t2≤Tc. Combining this with +(6.37), we infer that +t1(c4t1c−4)≤Tc. (6.38) +Using that c3t1c−3=t1, due to Lemma 29 with d= 6, we see that this equation is +equivalent with (6.35). Therefore, we are facing exactly the same e numeration problem +here as forp= 3m/2, and, consequently, the number of solutions to (6.38) is the same , +namely3m+2 +2, as required. +Finally, we turn to (6.32d). By Remark 3, the only choices for h2andm2to be +considered are h2= 6 andm2= 2, respectively h2=m2= 2. These correspond to the +choicesp= 3m/2, respectively p= 9m/2, all of which have already been discussed as +they do not belong to (6.32d). Hence, (3.3) must necessarily hold, a s required. +CaseG27.The degrees are 6 ,12,30, and hence we have +Catm(G27;q) =[30m+30]q[30m+12]q[30m+6]q +[30]q[12]q[6]q. +Letζbe a 14m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(G27;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (6.39a) +lim +q→ζCatm(G27;q) =5m+2 +2,ifζ=ζ12,ζ4,2|m, (6.39b) +lim +q→ζCatm(G27;q) = Catm(G27),ifζ=ζ6,ζ3,−1,1, (6.39c) +lim +q→ζCatm(G27;q) = 1,otherwise. (6.39d) +We must now prove that the left-handside of (3.3) in each case agre es with the values +exhibited in (6.39). The only cases not covered by Lemmas 27 and 28 a re the ones in +(6.39b) and (6.39d). +We first consider (6.39b). By Lemma 26, we are free to choose p= 5m/2 ifζ=ζ12, +respectively p= 15m/2 ifζ=ζ4. In both cases, mmust be divisible by 2. +We start with the case that p= 5m/2. From (5.1), we infer +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c3wm +2+1c−3,c3wm +2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm +2c−2/parenrightbig +. +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c3wm +2+ic−3, i= 1,2,...,m +2, (6.40a) +wi=c2wi−m +2c−2, i=m +2+1,m +2+2,...,m. (6.40b)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 77 +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +2such that +ℓT(wi) =ℓT(wi+m +2) = 1. +Writingt1,t2forwi,wi+m +2, respectively, the equations (6.40) reduce to +t1=c3t2c−3, (6.41a) +t2=c2t1c−2. (6.41b) +One of these equations is in fact superfluous: if we substitute (6.41 b) in (6.41a), then +we obtaint1=c5t1c−5which is automatically satisfied due to Lemma 29 with d= 6. +Since (w0;w1,...,w m)∈NCm(G27), we must have t1t2≤Tc. Combining this with +(6.41), we infer that +t1(c2t1c−2)≤Tc. (6.42) +With the help of CHEVIE, one obtains five solutions for t1in this equation: +t1∈/braceleftbig +[1],[2],[15],[16],[28]/bracerightbig +, +each of them giving rise to m/2 elements of Fix NCm(G27)(φp) sinceiranges from 1 to +m/2. Here we have used the short notation of CHEVIEreferring to the internal ordering +of the roots of G27inCHEVIE. +In total, we obtain 1 + 5m +2=5m+2 +2elements in Fix NCm(G27)(φp), which agrees with +the limit in (6.39b). +In the case that p= 15m/2, we infer from (5.1) that +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c8wm +2+1c−8,c8wm +2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm +2c−7/parenrightbig +. +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c8wm +2+ic−8, i= 1,2,...,m +2, (6.43a) +wi=c7wi−m +2c−7, i=m +2+1,m +2+2,...,m. (6.43b) +There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s +are equal to ε, or there is an iwith 1≤i≤m +2such that +ℓT(wi) =ℓT(wi+m +2) = 1. +Writingt1,t2forwi,wi+m +2, respectively, the equations (6.43) reduce to +t1=c8t2c−8, (6.44a) +t2=c7t1c−7. (6.44b) +One of these equations is in fact superfluous: if we substitute (6.44 b) in (6.44a), then +we obtaint1=c15t1c−15which is automatically satisfied due to Lemma 29 with d= 2. +Since (w0;w1,...,w m)∈NCm(G27), we must have t1t2≤Tc. Combining this with +(6.44), we infer that +t1(c7t1c−7)≤Tc. (6.45) +Using that c5t1c−5=t1, due to Lemma 29 with d= 6, we see that this equation is +equivalent with (6.42). Therefore, we are facing exactly the same e numeration problem +here as forp= 5m/2, and, consequently, the number of solutions to (6.45) is the same , +namely5m+2 +2, as required.78 C. KRATTENTHALER AND T. W. M ¨ULLER +Finally, we turn to (6.39d). By Remark 3, the only choices for h2andm2to be +considered are h2= 6 andm2= 3,h2= 6 andm2= 2,h2=m2= 3, respectively +h2=m2= 2. These correspond to the choices p= 5m/3, 5m/2, 10m/3, respectively +15m/2, out of which only p= 5m/3 andp= 10m/3 have not yet been discussed and +belong tothecurrent case. If p= 5m/3, thecorresponding actionof φpis givenby (6.6), +so that we have to solve for t1withℓT(t1) = 1 in the equation (6.9). A computation +with the help of CHEVIEfinds no solution. If p= 10m/3, the corresponding action of +φpis given by (6.11), so that we have to solve for t1withℓT(t1) in the equation (6.14). +Using that c5t1c−5=t1, due to Lemma 29 with d= 6, we see that this equation is +equivalent with the one in (6.9). Hence, in both cases, the left-hand side of (3.3) is +equal to 1, as required. +CaseG28=F4.The degrees are 2 ,6,8,12, and hence we have +Catm(F4;q) =[12m+12]q[12m+8]q[12m+6]q[12m+2]q +[12]q[8]q[6]q[2]q. +Letζbe a 12m-th root of unity. The following cases on the right-hand side of (3.3) +occur: +lim +q→ζCatm(F4;q) =m+1,ifζ=ζ12, (6.46a) +lim +q→ζCatm(F4;q) =3m+2 +2,ifζ=ζ8,2|m, (6.46b) +lim +q→ζCatm(F4;q) = (m+1)(2m+1),ifζ=ζ6,ζ3, (6.46c) +lim +q→ζCatm(F4;q) =(m+1)(3m+2) +2,ifζ=ζ4, (6.46d) +lim +q→ζCatm(F4;q) = Catm(F4),ifζ=−1 orζ= 1, (6.46e) +lim +q→ζCatm(F4;q) = 1,otherwise. (6.46f) +We must now prove that the left-handside of (3.3) in each case agre es with the values +exhibited in (6.46). The only cases not covered by Lemmas 27 and 28 a re the ones in +(6.46b) and (6.46f). By Lemma 26, we are free to choose p= 3m/2. In particular, m +must be divisible by 2. From (5.1), we infer +φp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (∗;c2wm +2+1c−2,c2wm +2+2c−2,...,c2wmc−2,cw1c−1,...,cw m +2c−1/parenrightbig +. +Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations +wi=c2wm +2+ic−2, i= 1,2,...,m +2, (6.47a) +wi=cwi−m +2c−1, i=m +2+1,m +2+2,...,m. (6.47b) +There are four distinct possibilities for choosing the wi’s, 1≤i≤m: +(i) all thewi’s are equal to ε(andw0=c), +(ii) there is an iwith 1≤i≤m +2such that +ℓT(wi) =ℓT(wi+m +2) = 2, (6.48a) +and all other wj’s are equal to ε,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 79 +(iii) there is an iwith 1≤i≤m +2such that +ℓT(wi) =ℓT(wi+m +2) = 1, (6.48b) +and the other wj’s, 1≤j≤m, are equal to ε, +(iv) there are i1andi2with 1≤i1|S2(W)|,/producttext +i∈S1(W)(mh+di)/producttext +i∈S2(W)di/producttext +i/∈S1(W)(1−ζdi−h) +/producttext +i/∈S2(W)(1−ζdi),if|S1(W)|=|S2(W)|. +(8.4) +Since, by Theorem 25, Catm(W;q) is a polynomial in q, the case |S1(W)|<|S2(W)| +cannot occur. +We claim that, for the case where |S1(W)|=|S2(W)|, the factors in the quotient of +products/producttext +i/∈S1(W)(1−ζdi−h)/producttext +i/∈S2(W)(1−ζdi) +cancel pairwise. If we assume the correctness of the claim, it is obv ious that we get +the same result if we replace ζbyζk, where gcd( k,(m+1)h/p) = 1, hence establishing +(8.2). +In order to see that our claim is indeed valid, we proceed in a case-by- case fash- +ion, making appeal to the classification of irreducible well-generated complex reflection +groups, which werecalled inSection2. Firstofall, since dn=h, thesetS1(W)isalways +non-empty as it contains the element n. Hence, if we want to have |S1(W)|=|S2(W)|, +the setS2(W) must be non-empty as well. In other words, the integer ( m+ 1)h/p +must divide at least one of the degrees d1,d2,...,d n. In particular, this implies that, +for each fixed reflection group Wof exceptional type, only a finite number of values of +(m+1)h/phas to be checked. Writing Mfor (m+1)h/p, what needs to be checked is +whether the multisets (that is, multiplicities of elements must be taken into account) +{(di−h) modM:i /∈S1(W)}and{dimodM:i /∈S2(W)} +are the same. Since, for a fixed irreducible well-generated complex r eflection group, +thereisonlyafinitenumber ofpossibilities for M, thisamountstoaroutineverification. +/square +Lemma 35. Letpbe a divisor of (m+ 1)h. Ifpis divisible by m+ 1, then(7.2)is +true. +Proof.According to (8.1), the action of ψponNCm(W) is described by +ψp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (cp/(m+1)w0c−p/(m+1);cp/(m+1)w1c−p/(m+1),...,cp/(m+1)wmc−p/(m+1)/parenrightbig +. +Hence, if (w0;w1,...,w m) is fixed by ψp, then each individual wimust be fixed under +conjugation by cp/(m+1). +Using the notation W′= Cent W(cp/(m+1)), the previous observation means that wi∈ +W′,i= 1,2,...,m. By the theorem of Springer cited in the proof of Lemma 27 and by +(5.5), the tuples ( w0;w1,...,w m) fixed byψpare in fact identical with the elements ofCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13 9 +NCm(W′), which implies that +|FixNCm(W)(ψp)|=|NCm(W′)|. (8.5) +Application of Theorem 1 with Wreplaced by W′and of the “limit rule” (5.4) then +yields that +|NCm(W′)|=/productdisplay +1≤i≤n +(m+1)h +p|dimh+di +di= Catm(W;q)/vextendsingle/vextendsingle +q=e2πip/(m+1)h. (8.6) +Combining (8.5) and (8.6), we obtain (7.2). This finishes the proof of t he lemma. /square +Lemma 36. Equation (7.2)holds for all divisors pofm+1. +Proof.We have +Catm(W;q)/vextendsingle/vextendsingle +q=e2πip/(m+1)h=/braceleftBigg +0 ifpnthen +FixNCm(W)(ψp) =∅. +Proof.Let us suppose that ( w0;w1,...,w m)∈FixNCm(W)(ψp) and that there exists a +j≥1 such that wj/ne}ationslash=ε. By (8.7), it then follows for such a jthat alsowk/ne}ationslash=εfor all +k≡j−lm1b(modm+ 1), where, as before, bis defined as the unique integer with +h1=am2+band 0≤b n, a contradiction. This +leaves as only possibility ( w0;w1,...,w m) = (c;ε,...,ε). However, this is clearly not +an element of Fix NCm(W)(ψp) unlesspis divisible by m+1. This is impossible since +p +m+1=m1h1 +m1m2=h1 +m2 +is not an integer by our hypotheses. /square +Remark 4.(1) If we put ourselves in the situation of the assumptions of Lemma 37, +then we may conclude that equation (7.2) only needs to be checked f or pairs (m2,h2) +subject to the following restrictions: +m2≥2,gcd(h1,m2) = 1,andh2divides all degrees of W. (8.8) +Indeed, Lemmas 35 and 37 together imply that equation (7.2) is alway s satisfied except +ifm2≥2,h2divides all degrees of W, and gcd(h1,m2) = 1. +(2) Still putting ourselves in the situation of Lemma 37, if m2> nandm2h2does +not divide any of the degrees of W, then equation (7.2) is satisfied. Indeed, Lemma 38 +says that in this case the left-hand side of (7.2) equals 0, while it is obv ious that in this +case the right-hand side of (7.2) equals 0 as well. +(3)It shouldbeobserved that thisleaves afinitenumber of choices form2to consider, +whence a finite number of choices for ( m1,m2,h1,h2). Altogether, there remains a finite +number of choices for p=h1m1to be checked.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 1 +Lemma 39. LetWbe an irreducible well-generated complex reflection group o f rankn +with the property that di|hfori= 1,2,...,n. Then Theorem 33is true for this group +W. +Proof.By Lemma 34, we may restrict ourselves to divisors pof (m+1)h. +Supposethat e2πip/(m+1)hisadi-throotofunityforsome i. Inotherwords, ( m+1)h/p +dividesdi. Sincediis a divisor of hby assumption, the integer ( m+1)h/palso divides +h. But this is equivalent to saying that m+ 1 divides p, and equation (7.2) holds by +Lemma 35. +Now assume that ( m+1)h/pdoes not divide any of the di’s. In this case, it follows +from (8.4) and the fact that we have S1(W)⊇ {n}andS2(W) =∅that the right- +hand side of (7.2) equals 0. Inspection of the classification of all irre ducible well- +generated complex reflection groups, which we recalled in Section 2, reveals that all +groups satisfying the hypotheses of the lemma have rank n≤2. Except for the groups +contained in the infinite series G(d,1,n) andG(e,e,n) for which Theorem 2 has been +established in [19], these are the groups G5,G6,G9,G10,G14,G17,G18,G21. We now +discuss these groups case by case, keeping the notation of Lemma 37. In order to +simplify the argument, we note that Lemma 38 implies that equation (7 .2) holds if +m2>2, so that in the following arguments we always may assume that m2= 2. +CaseG5. The degrees are 6 ,12, and therefore Remark 4.(1) implies that equa- +tion (7.2) is always satisfied. +CaseG6. The degrees are 4 ,12, and therefore, according to Remark 4.(1), we need +only consider the case where h2= 4 andm2= 2, that is, p= 3(m+1)/2. Then (8.7) +becomes +ψp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (c2wm+1 +2c−2;c2wm+3 +2c−2,...,c2wmc−2,cw0c−1,...,cw m−1 +2c−1/parenrightbig +. +(8.9) +If (w0;w1,...,w m) is fixed by ψp, there must exist an iwith 0≤i≤m−1 +2such that +ℓT(wi) = 1,wicwic−1=c, and allwj,j/ne}ationslash=i,m+1 +2+i, equalε. However, with the help of +CHEVIE, one verifies that there is no such solution to this equation. Hence, the left-hand +side of (7.2) is equal to 0, as required. +CaseG9. The degrees are 8 ,24, and therefore, according to Remark 4.(1), we need +only consider the case where h2= 8 andm2= 2, that is, p= 3(m+1)/2. This is the +samepas forG6. Again,CHEVIEfinds no solution. Hence, the left-hand side of (7.2) is +equal to 0, as required. +CaseG10. The degrees are 12 ,24, and therefore, according to Remark 4.(1), we need +only consider the case where h2= 12 andm2= 2, that is, p= 3(m+1)/2. This is the +samepas forG6. Again,CHEVIEfinds no solution. Hence, the left-hand side of (7.2) is +equal to 0, as required. +CaseG14. The degrees are 6 ,24, and therefore Remark 4.(1) implies that equa- +tion (7.2) is always satisfied. +CaseG17. The degrees are 20 ,60, and therefore, according to Remark 4.(1), we +need only consider the cases where h2= 20 andm2= 2, respectively that h2= 4 and +m2= 2. In the first case, p= 3(m+1)/2, which is the same pas forG6. Again,CHEVIE142 C. KRATTENTHALER AND T. W. M ¨ULLER +finds no solution. In the second case, p= 15(m+1)/2. Then (8.7) becomes +ψp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (c8wm+1 +2c−8;c8wm+3 +2c−8,...,c8wmc−8,c7w0c−7,...,c7wm−1 +2c−7/parenrightbig +.(8.10) +By Lemma 29, every element of NC(W) is fixed under conjugation by c3, and, thus, +on elements fixed by ψp, the above action of ψpreduces to the one in (8.9). This action +was already discussed in the first case. Hence, in both cases, the le ft-hand side of (7.2) +is equal to 0, as required. +CaseG18. The degrees are 30 ,60, and therefore Remark 4.(1) implies that equa- +tion (7.2) is always satisfied. +CaseG21. The degrees are 12 ,60, and therefore, according to Remark 4.(1), we +need only consider the cases where h2= 5 andm2= 2, respectively that h2= 15 and +m2= 2. In the first case, p= 5(m+1)/2, so that (8.7) becomes +ψp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (c3wm+1 +2c−3;c3wm+3 +2c−3,...,c3wmc−3,c2w0c−2,...,c2wm−1 +2c−2/parenrightbig +.(8.11) +If (w0;w1,...,w m) is fixed by ψp, there must exist an iwith 0≤i≤m−1 +2such that +ℓT(wi) = 1 and wic2wic−2=c. However, with the help of CHEVIE, one verifies that +there is no such solution to this equation. In the second case, p= 15(m+1)/2. Then +(8.7) becomes the action in (8.10). By Lemma 29, every element of NC(W) is fixed +under conjugation by c5, and, thus, on elements fixed by ψp, the action of ψpin (8.10) +reduces to the one in the first case. Hence, in both cases, the left -hand side of (7.2) is +equal to 0, as required. +This completes the proof of the lemma. /square +9.Case-by-case verification of Theorem 33 +We now perform a case-by-case verification of Theorem 33. It sho uld be observed +that theactionof ψ(givenin (7.1)) isexactly thesame astheactionof φ(given in(3.1)) +withmreplaced by m+1on the components w1,w2,...,w m+1, that is, if we disregard +the 0-th component of the elements of the generalised non-cross ing partitions involved. +The only difference which arises is that, while the ( m+ 1)-tuples ( w0;w1,...,w m) in +(7.1) must satisfy w0w1···wm=c, forw1,w2,...,w m+1in (3.1) we only must have +w1w2···wm+1≤Tc. The condition for ( w0;w1,...,w m) of being in Fix NCm(W)(ψp) +is therefore exactly the same as the condition on w1,w2,...,w m+1for the element +(ε;w1,...,w m,wm+1)beinginFix NCm+1(W)(φp). Consequently, wemayusethecounting +results from Section 6, except that we have to restrict our atten tion to those elements +(w0;w1,...,w m,wm+1)∈NCm+1(W) for which w1w2···wm+1=c, or, equivalently, +w0=ε. +As before, we write ζdfor a primitive d-th root of unity. +CaseG4.The degrees are 4 ,6, and hence we have +Catm(G4;q) =[6m+6]q[6m+4]q +[6]q[4]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 3 +Letζbe a 6(m+ 1)-th root of unity. As before, in what follows we abbreviate the +assertion that “ ζis a primitive d-th root of unity” as “ ζ=ζd.” The following cases on +the right-hand side of (7.2) occur: +lim +q→ζCatm(G4;q) =m+1,ifζ=ζ6,ζ3, (9.1a) +lim +q→ζCatm(G4;q) =3m+3 +2,ifζ=ζ4,2|(m+1), (9.1b) +lim +q→ζCatm(G4;q) = Catm(G4),ifζ=−1 orζ= 1, (9.1c) +lim +q→ζCatm(G4;q) = 0,otherwise. (9.1d) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.1). The only cases not covered by Lemmas 35 an d 36 are the +ones in (9.1b) and (9.1d). On the other hand, the only case left to co nsider according to +Remark 4 is the case where h2=m2= 2, that is the case (9.1b) where p= 3(m+1)/2. +In particular, m+ 1 must be divisible by 2. The action of ψpis the same as the one +in (8.9). Hence, the counting problem is the same as there, except t hat the underlying +group now is G4. With the help of CHEVIE, one finds that each of the 3 (complex) +reflections in G4which are less than the (chosen) Coxeter element is a valid choice for +wi, and each of these choices gives rise to ( m+1)/2 elements in Fix NCm(G4)(ψp) since +the indexiranges from 0 to ( m−1)/2. +Hence, in total, we obtain 3m+1 +2=3m+3 +2elements in Fix NCm(G4)(ψp), which agrees +with the limit in (9.1b). +CaseG8.The degrees are 8 ,12, and hence we have +Catm(G8;q) =[12m+12]q[12m+8]q +[12]q[8]q. +Letζbe a 12(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G8;q) =m+1,ifζ=ζ12,ζ6,ζ3, (9.2a) +lim +q→ζCatm(G8;q) =3m+3 +2,ifζ=ζ8,2|(m+1), (9.2b) +lim +q→ζCatm(G8;q) = Catm(G8),ifζ=ζ4,−1,1, (9.2c) +lim +q→ζCatm(G8;q) = 0,otherwise. (9.2d) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.2). The only cases not covered by Lemmas 35 an d 36 are the +ones in (9.2b) and (9.2d). On the other hand, the only case left to co nsider according +to Remark 4 is the case where h2= 4 andm2= 2, that is the case (9.2b) where +p= 3(m+ 1)/2. In particular, m+1 must be divisible by 2. The action of ψpis the +same as the one in (8.9). Hence, the counting problem is the same as t here, except +that the underlying group now is G8. With the help of CHEVIE, one finds that each +of the 3 (complex) reflections in G8which are less than the (chosen) Coxeter element +is a valid choice for wi, and each of these choices gives rise to ( m+ 1)/2 elements in +FixNCm(G8)(ψp) since the index iranges from 0 to ( m−1)/2.144 C. KRATTENTHALER AND T. W. M ¨ULLER +Hence, in total, we obtain 3m+1 +2=3m+3 +2elements in Fix NCm(G8)(ψp), which agrees +with the limit in (9.2b). +CaseG16.The degrees are 20 ,30, and hence we have +Catm(G16;q) =[30m+30]q[30m+20]q +[30]q[20]q. +Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G16;q) =m+1,ifζ=ζ30,ζ15,ζ6,ζ3, (9.3a) +lim +q→ζCatm(G16;q) =3m+3 +2,ifζ=ζ20,ζ4,2|(m+1), (9.3b) +lim +q→ζCatm(G16;q) = Catm(G16),ifζ=ζ10,ζ5,−1,1, (9.3c) +lim +q→ζCatm(G16;q) = 0,otherwise. (9.3d) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.3). The only cases not covered by Lemmas 35 an d 36 are the +ones in (9.3b) and (9.3d). On the other hand, the only cases left to c onsider according +to Remark 4 are the cases where h2= 10 andm2= 2, respectively h2=m2= 2. Both +cases belong to (9.3b). In the first case, we have p= 3(m+1)/2, while in the second +case we have p= 15(m+ 1)/2. In particular, m+ 1 must be divisible by 2. In the +first case, the action of ψpis the same as the one in (8.9). Hence, the counting problem +is the same as there, except that the underlying group now is G16. With the help of +CHEVIE, one finds that each of the 3 (complex) reflections in G16which are less than the +(chosen) Coxeter element is a valid choice for wi, and each of these choices gives rise to +(m+ 1)/2 elements in Fix NCm(G16)(ψp) since the index iranges from 0 to ( m−1)/2. +On the other hand, if p= 15(m+1)/2, then the action of ψpis the same as the one in +(8.10). By Lemma 29, every element of NC(G16) is fixed under conjugation by c3, and, +thus, on elements fixed by ψp, the action of ψpreduces to the one in the first case. +Hence, in total, we obtain 3m+1 +2=3m+3 +2elements in Fix NCm(G16)(ψp), which agrees +with the limit in (9.3b). +CaseG20.The degrees are 12 ,30, and hence we have +Catm(G20;q) =[30m+30]q[30m+12]q +[30]q[12]q. +Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G20;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (9.4a) +lim +q→ζCatm(G20;q) =5m+5 +2,ifζ=ζ12,ζ4,2|(m+1), (9.4b) +lim +q→ζCatm(G20;q) = Catm(G20),ifζ=ζ6,ζ3,−1,1, (9.4c) +lim +q→ζCatm(G20;q) = 0,otherwise. (9.4d)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 5 +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.4). The only cases not covered by Lemmas 35 an d 36 are the +ones in (9.4b) and (9.4d). On the other hand, the only cases left to c onsider according +to Remark 4 are the cases where h2= 6 andm2= 2, respectively h2=m2= 2. Both +cases belong to (9.4b). In the first case, we have p= 5(m+1)/2, while in the second +case we have p= 15(m+1)/2. In particular, m+1 must be divisible by 2. In the first +case, the action of ψpis the same as the one in (8.11). Hence, the counting problem +is the same as there, except that the underlying group now is G20. With the help of +CHEVIE, one finds that each of the 5 (complex) reflections in G20which are less than the +(chosen) Coxeter element is a valid choice for wi, and each of these choices gives rise to +(m+ 1)/2 elements in Fix NCm(G20)(ψp) since the index iranges from 0 to ( m−1)/2. +On the other hand, if p= 15(m+1)/2, then the action of ψpis the same as the one in +(8.10). By Lemma 29, every element of NC(G20) is fixed under conjugation by c5, and, +thus, on elements fixed by ψp, the action of ψpreduces to the one in the first case. +Hence, in total, we obtain 5m+1 +2=5m+5 +2elements in Fix NCm(G20)(ψp), which agrees +with the limit in (9.4b). +CaseG23=H3.The degrees are 2 ,6,10, and hence we have +Catm(H3;q) =[10m+10]q[10m+6]q[10m+2]q +[10]q[6]q[2]q. +Letζbe a 10(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(H3;q) =m+1,ifζ=ζ10,ζ5, (9.5a) +lim +q→ζCatm(H3;q) =5m+5 +3,ifζ=ζ6,ζ3,3|(m+1), (9.5b) +lim +q→ζCatm(H3;q) = Catm(H3),ifζ=−1 orζ= 1, (9.5c) +lim +q→ζCatm(H3;q) = 0,otherwise. (9.5d) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.5). The only cases not covered by Lemmas 35 an d 36 are the +ones in (9.5b) and (9.5d). On the other hand, the only cases left to c onsider according +to Remark 4 are the cases where h2= 1 andm2= 3,h2= 2 andm2= 3, and +h2=m2= 2. These correspond to the choices p= 10(m+ 1)/3,p= 5(m+ 1)/3, +respectively p= 5(m+1)/2. The first two cases belong to (9.5b), while p= 5(m+1)/2 +belongs to (9.5d). +In the case that p= 5(m+1)/3, the action of ψpis given by +ψp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (c2wm+1 +3c−2;c2wm+4 +3c−2,...,c2wmc−2,cw0c−1,...,cw m−2 +3c−1/parenrightbig +. +Hence, for an iwith 0≤i≤m−2 +3, we must find an element wi=t1, wheret1satisfies +(6.9), andall other wj,j /∈/braceleftbig +i,i+m+1 +3,i+2(m+1) +3/bracerightbig +, are set equal to ε. We have found five +solutions to the counting problem (6.9) in (6.10). Each of them gives r ise to (m+1)/3 +elements in Fix NCm(H3)(ψp) since the index iranges from 0 to ( m−2)/3. On the other146 C. KRATTENTHALER AND T. W. M ¨ULLER +hand, ifp= 10(m+1)/3, then the action of ψpis given by +ψp/parenleftbig +(w0;w1,...,w m)/parenrightbig += (c4w2m+2 +3c−4;c4w2m+5 +3c−4,...,c4wmc−4,c3w0c−3,...,c3w2m−1 +3c−3/parenrightbig +. +By Lemma 29, every element of NC(H3) is fixed under conjugation by c5, and, thus, +on elements fixed by ψp, the action of ψpreduces to the one in the first case. +Hence, in total, we obtain 5m+1 +3=5m+5 +3elements in Fix NCm(H3)(ψp), which agrees +with the limit in (9.5b). +Ifp= 5(m+ 1)/2, then the action of ψpis the same as the one in (8.11). The +computation at the end of Case H3in Section 6 did not find any solutions, which is in +agreement with (9.5d). +CaseG24.The degrees are 4 ,6,14, and hence we have +Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q +[14]q[6]q[4]q. +Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (9.6a) +lim +q→ζCatm(G24;q) =7m+7 +3,ifζ=ζ6,ζ3,3|(m+1), (9.6b) +lim +q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (9.6c) +lim +q→ζCatm(G24;q) = 0,otherwise. (9.6d) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.6). The only cases not covered by Lemmas 35 an d 36 are the +ones in (9.6b) and (9.6d). On the other hand, the only cases left to c onsider according +to Remark 4 are the cases where h2= 1 andm2= 3,h2= 2 andm2= 3, and +h2=m2= 2. These correspond to the choices p= 14(m+ 1)/3,p= 7(m+ 1)/3, +respectively p= 7(m+1)/2. The first two cases belong to (9.6b), while p= 7(m+1)/2 +belongs to (9.6d). +In the case that p= 14(m+1)/3 orp= 7(m+1)/3, we have found seven solutions to +the counting problem (6.18) in (6.19), and each of them gives rise to ( m+1)/3 elements +in Fix NCm(G24)(ψp) (in the style as discussed in Case H3). Hence, in total, we obtain +7m+1 +3=7m+7 +3elements in Fix NCm(G24)(ψp), which agrees with the limit in (9.6b). +Ifp= 7(m+ 1)/2, the relevant counting problem is (6.25). However, no element +(w0;w1,...,w m)∈FixNCm(G24)(ψp) can be produced in this way since the counting +problem imposes the restriction that ℓT(w0) +ℓT(w1) +···+ℓT(wm) be even, which +contradicts the fact that ℓT(c) =n= 3. This is in agreement with the limit in (9.6d). +CaseG25.The degrees are 6 ,9,12, and hence we have +Catm(G25;q) =[12m+12]q[12m+9]q[12m+6]q +[12]q[9]q[6]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 7 +Letζbe a 12(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G25;q) =m+1,ifζ=ζ12,ζ4, (9.7a) +lim +q→ζCatm(G25;q) =4m+4 +3,ifζ=ζ9,3|(m+1), (9.7b) +lim +q→ζCatm(G25;q) = (m+1)(2m+1),ifζ=ζ6,−1 (9.7c) +lim +q→ζCatm(G25;q) = Catm(G25),ifζ=ζ3,1, (9.7d) +lim +q→ζCatm(G25;q) = 0,otherwise. (9.7e) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.7). The only cases not covered by Lemmas 35 an d 36 are the +ones in (9.7b) and (9.7e). On the other hand, the only case left to co nsider according to +Remark4isthecasewhere h2=m2= 3. Thiscorrespondstothechoice p= 4(m+1)/3, +which belongs to (9.7b). We have found four solutions to the countin g problem (6.30) +in (6.31), and each of them gives rise to ( m+1)/3 elements in Fix NCm(G25)(ψp) (in the +style as discussed in Case H3). Hence, in total, we obtain 4m+1 +3=4m+4 +3elements in +FixNCm(G25)(ψp), which agrees with the limit in (9.7b). +CaseG26.The degrees are 6 ,12,18, and hence we have +Catm(G26;q) =[18m+18]q[18m+12]q[18m+6]q +[18]q[12]q[6]q. +Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G26;q) =m+1,ifζ=ζ18,ζ9, (9.8a) +lim +q→ζCatm(G26;q) = Catm(G26),ifζ=ζ6,ζ3,−1,1, (9.8b) +lim +q→ζCatm(G26;q) = 0,otherwise. (9.8c) +We must now prove that the left-handside of (7.2) in each case agre es with the values +exhibited in (9.8). The only case not covered by Lemmas 35 and 36 is th e one in (9.8c). +On the other hand, the only cases left to consider according to Rem ark 4 are the cases +whereh2= 6 andm2= 2, respectively h2=m2= 2. These correspond to the choices +p= 3(m+1)/2,respectively p= 9(m+1)/2,bothofwhichbelongto(9.8c). Therelevant +counting problem is (6.35). However, no element ( w0;w1,...,w m)∈FixNCm(G26)(ψp) +can be produced in this way since the counting problem imposes the re striction that +ℓT(w0)+ℓT(w1)+···+ℓT(wm) be even, which is absurd. This is in agreement with the +limit in (9.8c). +CaseG27.The degrees are 6 ,12,30, and hence we have +Catm(G27;q) =[30m+30]q[30m+12]q[30m+6]q +[30]q[12]q[6]q.148 C. KRATTENTHALER AND T. W. M ¨ULLER +Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G27;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (9.9a) +lim +q→ζCatm(G27;q) = Catm(G27),ifζ=ζ6,ζ3,−1,1, (9.9b) +lim +q→ζCatm(G27;q) = 0,otherwise. (9.9c) +We must now prove that the left-handside of (7.2) in each case agre es with the values +exhibited in (9.9). The only case not covered by Lemmas 35 and 36 is th e one in (9.9c). +On the other hand, the only cases left to consider according to Rem ark 4 are the cases +whereh2= 6 andm2= 3,h2=m2= 3,h2= 6 andm2= 2, respectively h2=m2= 2. +These correspond to the choices p= 5(m+1)/3, 10(m+1)/3, 5(m+1)/2, respectively +15(m+1)/2, all of which belong to (9.9c). +Ifp= 5(m+ 1)/3 orp= 10(m+ 1)/3, the computation with the help of CHEVIE +at the end of Case G27in Section 6 did not find any solutions for the corresponding +counting problem. This is in agreement with the limit in (9.9c). +In the case that 5( m+1)/2 or 15(m+1)/2, the relevant counting problem is (6.42). +However, no element ( w0;w1,...,w m)∈FixNCm(G27)(ψp) can be produced in this way +since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+···+ℓT(wm) +be even, which is absurd. This is again in agreement with the limit in (9.9c) . +CaseG28=F4.The degrees are 2 ,6,8,12, and hence we have +Catm(F4;q) =[12m+12]q[12m+8]q[12m+6]q[12m+2]q +[12]q[8]q[6]q[2]q. +Letζbe a 12(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(F4;q) =m+1,ifζ=ζ12, (9.10a) +lim +q→ζCatm(F4;q) =3m+3 +2,ifζ=ζ8,2|(m+1), (9.10b) +lim +q→ζCatm(F4;q) = (m+1)(2m+1),ifζ=ζ6,ζ3, (9.10c) +lim +q→ζCatm(F4;q) =(m+1)(3m+2) +2,ifζ=ζ4, (9.10d) +lim +q→ζCatm(F4;q) = Catm(F4),ifζ=−1 orζ= 1, (9.10e) +lim +q→ζCatm(F4;q) = 0,otherwise. (9.10f) +We must now prove that the left-handside of (7.2) in each case agre es with the values +exhibited in (9.10). The only cases not covered by Lemmas 35 and 36 a re the ones in +(9.10b) and (9.10f). On the other hand, according to Remark 4, th e are no choices for +h2andm2left to be considered. +CaseG29.The degrees are 4 ,8,12,20, and hence we have +Catm(G29;q) =[20m+20]q[20m+12]q[20m+8]q[20m+4]q +[20]q[12]q[8]q[4]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 9 +Letζbe a 20(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G29;q) =m+1,ifζ=ζ20,ζ10,ζ5, (9.11a) +lim +q→ζCatm(G29;q) = Catm(G29),ifζ=ζ4,−1,1, (9.11b) +lim +q→ζCatm(G29;q) = 0,otherwise. (9.11c) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.11). The only case not covered by Lemmas 35 an d 36 is the one +in(9.11c). Ontheother hand, theonly cases left to consider accor ding to Remark4, the +only choices for h2andm2to be considered are h2= 1 andm2= 3,h2= 2 andm2= 3, +h2= 4 andm2= 3,h2= 4 andm2= 2, respectively h2=m2= 4. These correspond +to the choices p= 20(m+ 1)/3,p= 10(m+ 1)/3,p= 5(m+ 1)/3,p= 5(m+ 1)/2, +respectively p= 5(m+1)/4, all of which belong to (9.11c). +In the case that p= 20(m+1)/3,p= 10(m+1)/3, orp= 5(m+1)/3, the relevant +counting problem is (6.55). However, no element ( w0;w1,...,w m)∈FixNCm(G27)(ψp) +can be produced in this way since the counting problem imposes the re striction that +ℓT(w0)+ℓT(w1)+···+ℓT(wm) be divisible by 3, which is absurd. This is in agreement +with the limit in (9.11c). +In the case that p= 5(m+1)/2, the relevant counting problem is (6.64), for which +we did not find any solutions. This is again in agreement with the limit in (9.1 1c). +In the case that p= 5(m+1)/4, the computation at the end of Case G29in Section 6 +did not find any solutions, which is as well in agreement with the limit in (9.1 1c). +CaseG30=H4.The degrees are 2 ,12,20,30, and hence we have +Catm(H4;q) =[30m+30]q[30m+20]q[30m+12]q[30m+2]q +[30]q[20]q[12]q[2]q. +Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(H4;q) =m+1,ifζ=ζ30,ζ15, (9.12a) +lim +q→ζCatm(H4;q) =3m+3 +2,ifζ=ζ20,2|(m+1), (9.12b) +lim +q→ζCatm(H4;q) =5m+5 +2,ifζ=ζ12,2|(m+1), (9.12c) +lim +q→ζCatm(H4;q) =(m+1)(3m+2) +2,ifζ=ζ10,ζ5, (9.12d) +lim +q→ζCatm(H4;q) =(m+1)(5m+2) +2,ifζ=ζ6,ζ3, (9.12e) +lim +q→ζCatm(H4;q) =(m+1)(15m+1) +4,ifζ=ζ4,2|(m+1), (9.12f) +lim +q→ζCatm(H4;q) = Catm(H4),ifζ=−1 orζ= 1, (9.12g) +lim +q→ζCatm(H4;q) = 0,otherwise. (9.12h) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.12). The only cases not covered by Lemmas 35 a nd 36 are150 C. KRATTENTHALER AND T. W. M ¨ULLER +the ones in (9.12b), (9.12c), (9.12f), and (9.12h). On the other ha nd, the only cases +left to consider according to Remark 4 are the cases where h2= 2 andm2= 4, +respectively h2=m2= 2. Thesecorrespondtothechoices p= 15(m+1)/2,respectively +p= 15(m+1)/4, out of which the first belongs to (9.12f), while the second belongs to +(9.12h). +In the case that p= 15(m+1)/2, the action of ψpis the same as the one in (8.10). +We have found eight solutions to the counting problem (6.80) in (6.83) , each of them +giving rise to ( m+1)/2 elements in Fix NCm(H4)(ψp) since the index i(in (6.80)) ranges +from 0 to ( m−1)/2, and we have found 30 solutions to the counting problem (6.82) +in (6.84), each of them giving rise to/parenleftbig(m+1)/2 +2/parenrightbig +elements in Fix NCm(H4)(ψp) since 0 ≤ +i1< i2≤(m−1)/2 (in (6.82)). Hence, we obtain 8m+1 +2+ 30/parenleftbig(m+1)/2 +2/parenrightbig +=(m+1)(15m+1) +4 +elements in Fix NCm(H4)(ψp), which agrees with the limit in (9.12f). +Ifp= 15(m+1)/4, the computation at the end of Case H4in Section 6 did not find +any solutions, which is in agreement with the limit in (9.12h). +CaseG32.The degrees are 12 ,18,24,30, and hence we have +Catm(G32;q) =[30m+30]q[30m+24]q[30m+18]q[30m+12]q +[30]q[24]q[18]q[12]q. +Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G32;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (9.13a) +lim +q→ζCatm(G32;q) =5m+5 +4,ifζ=ζ24,ζ8,4|(m+1), (9.13b) +lim +q→ζCatm(G32;q) =(5m+5)(5m+3) +8,ifζ=ζ12,ζ4,2|(m+1), (9.13c) +lim +q→ζCatm(G32;q) = Catm(G32),ifζ=ζ6,ζ3,−1,1, (9.13d) +lim +q→ζCatm(G32;q) = 0,otherwise. (9.13e) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.13). The only cases not covered by Lemmas 35 a nd 36 are the +ones in (9.13b), (9.13c), and (9.13e). On the other hand, the only c ases left to consider +according to Remark 4 are the cases where h2= 2 andm2= 4,h2= 6 andm2= 4, +h2=m2= 3,h2= 6 andm2= 3,h2=m2= 2, respectively h2= 6 andm2= 2. +These correspond to the choices p= 15(m+1)/4,p= 5(m+1)/4,p= 10(m+ 1)/3, +p= 5(m+1)/3,p= 15(m+1)/2, respectively p= 5(m+1)/2, out of which the first two +belong to (9.13b), the next two belong to (9.13e), and the last two b elong to (9.13c). +In the case that p= 15(m+1)/4 orp= 5(m+1)/4, we have found five solutions to +the counting problem (6.88) in (6.89), each of them giving rise to ( m+1)/4 elements +in Fix NCm(G32)(ψp). Hence, we obtain 5m+1 +4=5m+5 +4elements in Fix NCm(G32)(ψp), which +agrees with the limit in (9.13b). +In the case that p= 10(m+1)/3 orp= 5(m+1)/3, the relevant counting problem +is (6.95). However, no element ( w0;w1,...,w m)∈FixNCm(G32)(ψp) can be produced +in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+ +···+ℓT(wm) be divisible by 3, which is absurd. This is in agreement with the limit in +(9.13e).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15 1 +In the case that p= 15(m+1)/2 orp= 5(m+1)/2, we have found ten solutions to +the counting problem (6.102) in (6.105), each of them giving rise to ( m+1)/2 elements +in Fix NCm(G32)(ψp), and we have found 25 solutions to the counting problem (6.104) in +(6.106), each of them giving rise to/parenleftbig(m+1)/2 +2/parenrightbig +elements in Fix NCm(G32)(ψp). Hence, we +obtain 10m+1 +2+ 25/parenleftbig(m+1)/2 +2/parenrightbig +=(5m+5)(5m+3) +8elements in Fix NCm(G32)(ψp), which agrees +with the limit in (9.13c). +CaseG33.The degrees are 4 ,6,10,12,18, and hence we have +Catm(G33;q) =[18m+18]q[18m+12]q[18m+10]q[18m+6]q[18m+4]q +[18]q[12]q[10]q[6]q[4]q. +Letζbe a 18(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G33;q) =m+1,ifζ=ζ18,ζ9, (9.14a) +lim +q→ζCatm(G33;q) =9m+9 +5,ifζ=ζ10,ζ5,5|(m+1), (9.14b) +lim +q→ζCatm(G33;q) =(m+1)(3m+2)(3m+1) +2,ifζ=ζ6,ζ3, (9.14c) +lim +q→ζCatm(G33;q) = Catm(G33),ifζ=−1 orζ= 1, (9.14d) +lim +q→ζCatm(G33;q) = 0,otherwise. (9.14e) +We must now prove that the left-handside of (7.2) in each case agre es with the values +exhibited in (9.14). The only cases not covered by Lemmas 35 and 36 a re the ones in +(9.14b) and (9.14e). On the other hand, the only cases left to cons ider according to +Remark 4 are the cases where h2= 1 andm2= 5,h2= 2 andm2= 5,h2= 2 and +m2= 4, respectively h2=m2= 2. These correspond to the choices p= 18(m+1)/5, +p= 9(m+ 1)/5,p= 9(m+ 1)/4, respectively p= 9(m+ 1)/2, out of which the first +two belong to (9.14b), while the others belong to (9.14e). +In the case that p= 18(m+1)/5 orp= 9(m+1)/5, we have found nine solutions +to the counting problem (6.115) in (6.116). Hence, we obtain 9m+1 +5=9m+9 +5elements in +FixNCm(G33)(ψp), which agrees with the limit in (9.14b). +Ifp= 9(m+1)/4, the computation at the end of Case G33in Section 6 did not find +any solutions, which is again in agreement with the limit in (9.13e). +In the case that p= 9(m+ 1)/2, the relevant counting problems are (6.122) and +(6.124). However, no element ( w0;w1,...,w m)∈FixNCm(G33)(ψp) can be produced in +this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+···+ +ℓT(wm) be even, which is absurd. This is in agreement with the limit in (9.14e). +CaseG34.The degrees are 6 ,12,18,24,30,42, and hence we have +Catm(G34;q) =[42m+42]q[42m+30]q[42m+24]q +[42]q[30]q[24]q +×[42m+18]q[42m+12]q[42m+6]q +[18]q[12]q[6]q.152 C. KRATTENTHALER AND T. W. M ¨ULLER +Letζbe a 42(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(G34;q) =m+1,ifζ=ζ42,ζ21,ζ14,ζ7, (9.15a) +lim +q→ζCatm(G34;q) = Catm(G34),ifζ=ζ6,ζ3,−1,1, (9.15b) +lim +q→ζCatm(G34;q) = 0,otherwise. (9.15c) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.15). The only case not covered by Lemmas 35 an d 36 is the one +in (9.15c). On the other hand, the only cases left to consider accor ding to Remark 4 +are the cases where h2= 1 andm2= 5,h2= 2 andm2= 5,h2= 3 andm2= 5,h2= 6 +andm2= 5,h2= 2 andm2= 4,h2= 6 andm2= 4,h2=m2= 3,h2= 6 andm2= 3, +h2=m2= 2,h2= 6 andm2= 2, respectively h2= 6 andm2= 6. These correspond +to the choices p= 42(m+1)/5,p= 21(m+1)/5,p= 14(m+1)/5,p= 7(m+ 1)/5, +p= 21(m+1)/4,p= 7(m+1)/4,p= 14(m+1)/3,p= 7(m+1)/3,p= 21(m+1)/2, +p= 7(m+1)/2, respectively p= 7(m+1)/6, all of which belong to (9.15c). +Inthecasethat p= 42(m+1)/5,p= 21(m+1)/5,p= 14(m+1)/5,orp= 7(m+1)/5, +the relevant counting problem is (6.128). However, no element ( w0;w1,...,w m)∈ +FixNCm(G34)(ψp) can be produced in this way since the counting problem imposes the +restriction that ℓT(w0)+ℓT(w1)+···+ℓT(wm) be divisible by 5, which is absurd. This +is in agreement with the limit in (9.15c). +In the case that p= 21(m+1)/4 orp= 7(m+1)/4, the relevant counting problem +is (6.140). However, no element ( w0;w1,...,w m)∈FixNCm(G34)(ψp) can be produced +in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+ +···+ℓT(wm) be divisible by 4, which is absurd. This is in agreement with the limit in +(9.15c). +In the case that p= 14(m+1)/3 orp= 7(m+1)/3, the relevant counting problems +are (6.146) and (6.148). However, the computations with the help o fCHEVIEperformed +in CaseG34in Section 6 did not find any solutions for (6.146) or (6.148). This is in +agreement with the limit in (9.15c). +In the case that p= 21(m+1)/2, the relevant counting problems are (6.157), (6.158), +and (6.159). However, the computations with the help of CHEVIEperformed in Case G34 +in Section 6 found no wiwithℓT(wi) = 3 in (6.157), and hence no solutions for( wi1,wi2) +withℓT(wi1)+ℓT(wi2) = 3 in (6.158), and no solutions for ( wi1,wi2,wi3) in (6.159). This +is in agreement with the limit in (9.15c). +Ifp= 7(m+1)/6, the computation at the end of Case G34in Section 6 did not find +any solutions, which is also in agreement with the limit in (9.15c). +CaseG35=E6.The degrees are 2 ,5,6,8,9,12, and hence we have +Catm(E6;q) =[12m+12]q[12m+9]q[12m+8]q[12m+6]q[12m+5]q[12m+2]q +[12]q[9]q[8]q[6]q[5]q[2]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15 3 +Letζbe a 12(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(E6;q) =m+1,ifζ=ζ12, (9.16a) +lim +q→ζCatm(E6;q) =4m+4 +3,ifζ=ζ9,3|(m+1), (9.16b) +lim +q→ζCatm(E6;q) =3m+3 +2,ifζ=ζ8,2|(m+1), (9.16c) +lim +q→ζCatm(E6;q) = (m+1)(2m+1),ifζ=ζ6, (9.16d) +lim +q→ζCatm(E6;q) =(m+1)(3m+2) +2,ifζ=ζ4, (9.16e) +lim +q→ζCatm(E6;q) =(m+1)(4m+3)(2m+1) +3,ifζ=ζ3, (9.16f) +lim +q→ζCatm(E6;q) =(m+1)(3m+2)(2m+1)(6m+1) +2,ifζ=−1, (9.16g) +lim +q→ζCatm(E6;q) = Catm(E6),ifζ= 1, (9.16h) +lim +q→ζCatm(E6;q) = 0,otherwise. (9.16i) +We must now prove that the left-hand side of (7.2) in each case agre es with the +values exhibited in (9.16). The only cases not covered by Lemmas 35 a nd 36 are the +ones in (9.16b), (9.16c), and (9.16i). On the other hand, the only c ases left to consider +according to Remark 4 are the cases where h2= 1 andm2= 5, respectively h2= 2 and +m2= 5. These correspond to the choices p= 12(m+1)/5, respectively p= 6(m+1)/5, +both of which belong to (9.16i). +In the case that p= 12(m+1)/5, the relevant counting problem is (6.172). However, +no element ( w0;w1,...,w m)∈FixNCm(E6)(ψp) can be produced in this way since the +counting problemimposes therestriction that ℓT(w0)+ℓT(w1)+···+ℓT(wm)bedivisible +by 5, which is absurd. This is in agreement with the limit in (9.16i). +Ifp= 6(m+1)/5, the computation at the end of Case E6in Section 6 did not find +any solutions, which is also in agreement with the limit in (9.16i). +CaseG36=E7.The degrees are 2 ,6,8,10,12,14,18, and hence we have +Catm(E7;q) =[18m+18]q[18m+14]q[18m+12]q +[18]q[14]q[12]q +×[18m+10]q[18m+8]q[18m+6]q[18m+2]q +[10]q[8]q[6]q[2]q.154 C. KRATTENTHALER AND T. W. M ¨ULLER +Letζbe a 18(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(E7;q) =m+1,ifζ=ζ18,ζ9, (9.17a) +lim +q→ζCatm(E7;q) =9m+9 +7,ifζ=ζ14,ζ7,7|(m+1), (9.17b) +lim +q→ζCatm(E7;q) =(m+1)(3m+2)(3m+1) +2,ifζ=ζ6,ζ3, (9.17c) +lim +q→ζCatm(E7;q) = Catm(E7),ifζ=−1 orζ= 1, (9.17d) +lim +q→ζCatm(E7;q) = 0,otherwise. (9.17e) +We must now prove that the left-handside of (7.2) in each case agre es with the values +exhibited in (9.17). The only cases not covered by Lemmas 35 and 36 a re the ones in +(9.17b) and (9.17e). On the other hand, the only cases left to cons ider according to +Remark 4 are the cases where h2= 1 andm2= 7,h2= 2 andm2= 7,h2= 1 +andm2= 5,h2= 2 andm2= 5,h2= 2 andm2= 4, respectively h2=m2= 2. +These correspond to the choices p= 18(m+1)/7,p= 9(m+1)/7,p= 18(m+ 1)/5, +p= 9(m+ 1)/5,p= 9(m+ 1)/4, respectively p= 9(m+ 1)/2, out of which the first +two belong to (9.16b), and all others belong to (9.16i). +In the case that p= 18(m+1)/7 orp= 9(m+1)/7, we have found nine solutions +to the counting problem (6.176) in (6.177). Hence, we obtain 9m+1 +7=9m+9 +7elements in +FixNCm(E7)(ψp), which agrees with the limit in (9.17b). +In the case that p= 18(m+1)/5 orp= 9(m+1)/5, the relevant counting problem +is (6.186). However, no element ( w0;w1,...,w m)∈FixNCm(E7)(ψp) can be produced +in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+ +···+ℓT(wm) be divisible by 5, which is absurd. This is in agreement with the limit in +(9.17e). +In the case that p= 9(m+1)/4, the relevant counting problem is (6.192). However, +no element ( w0;w1,...,w m)∈FixNCm(E7)(ψp) can be produced in this way since the +counting problemimposes therestriction that ℓT(w0)+ℓT(w1)+···+ℓT(wm)bedivisible +by 4, which is absurd. This is again in agreement with the limit in (9.17e). +In the case that p= 9(m+1)/2, the relevant counting problems are (6.195), (6.196), +and (6.197). However, no element ( w0;w1,...,w m)∈FixNCm(E7)(ψp) can be produced +in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+ +···+ℓT(wm) be even, which is absurd. This is also in agreement with the limit in +(9.17e). +CaseG37=E8.The degrees are 2 ,8,12,14,18,20,24,30, and hence we have +Catm(E8;q) =[30m+30]q[30m+24]q[30m+20]q[30m+18]q +[30]q[24]q[20]q[18]q +×[30m+14]q[30m+12]q[30m+8]q[30m+2]q +[14]q[12]q[8]q[2]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15 5 +Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of +(7.2) occur: +lim +q→ζCatm(E8;q) =m+1,ifζ=ζ30,ζ15, (9.18a) +lim +q→ζCatm(E8;q) =5m+5 +4,ifζ=ζ24,4|(m+1), (9.18b) +lim +q→ζCatm(E8;q) =3m+3 +2,ifζ=ζ20,2|(m+1), (9.18c) +lim +q→ζCatm(E8;q) =(5m+5)(5m+3) +8,ifζ=ζ12,2|(m+1), (9.18d) +lim +q→ζCatm(E8;q) =(m+1)(3m+2) +2,ifζ=ζ10,ζ5, (9.18e) +lim +q→ζCatm(E8;q) =(5m+5)(15m+7) +16,ifζ=ζ8,4|(m+1), (9.18f) +lim +q→ζCatm(E8;q) =(m+1)(5m+4)(5m+3)(5m+2) +24,ifζ=ζ6,ζ3,(9.18g) +lim +q→ζCatm(E8;q) =(m+1)(5m+3)(15m+7)(15m+1) +64,ifζ=ζ4,2|(m+1), +(9.18h) +lim +q→ζCatm(E8;q) = Catm(E8),ifζ=−1 orζ= 1, (9.18i) +lim +q→ζCatm(E8;q) = 0,otherwise. (9.18j) +We must now prove that the left-handside of (7.2) in each case agre es with the values +exhibited in (9.18). The only cases not covered by Lemmas 35 and 36 a re the ones in +(9.18b), (9.18c), (9.18d), (9.18f), (9.18h), and (9.18j). On the o ther hand, the only +cases left to consider according to Remark 4 are the cases where h2= 2 andm2= 8, +h2= 1 andm2= 7,h2= 2 andm2= 7,h2= 2 andm2= 4, respectively h2=m2= 2. +These correspond to the choices p= 15(m+1)/8,p= 30(m+1)/7,p= 15(m+1)/7, +p= 15(m+1)/4, respectively 15( m+1)/2, out of which the first three belong to (9.18j), +the fourth belongs to (9.18f), and the last belongs to (9.18h). +Ifp= 15(m+1)/8, the relevant counting problem is (6.242). However, the computa - +tion at the end of Case E8in Section 6 did not find any solutions, which is in agreement +with the limit in (9.18j). Hence, the left-hand side of (7.2) is equal to 0 , as required. +In the case that p= 30(m+1)/7 orp= 15(m+1)/7, the relevant counting problem +is (6.213). However, no element ( w0;w1,...,w m)∈FixNCm(E8)(ψp) can be produced +in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+ +···+ℓT(wm) be divisible by 7, which is absurd. This is also in agreement with the limit +in (9.18j). +In the case that p= 15(m+ 1)/4, the relevant counting problems are (6.227) and +(6.228). We have found 45 solutions wito (6.227) of type A2 +1in (6.229), and we have +found 20 solutions wito (6.227) of type A2in (6.230), which implied 150 solutions for +(wi1,wi2) to (6.228). The first two give rise to to (45 + 20)m+1 +4= 65m+1 +4elements in +FixNCm(E8)(ψp), while the third give rise to 150/parenleftbig(m+1)/4 +2/parenrightbig +elements in Fix NCm(E8)(ψp). +Hence, we obtain 65m+1 +4+ 150/parenleftbig(m+1)/4 +2/parenrightbig +=(5m+5)(15m+7) +16elements in Fix NCm(E8)(ψp), +which agrees with the limit in (9.18f).156 C. KRATTENTHALER AND T. W. M ¨ULLER +In the case that p= 15(m+1)/2, the relevant counting problems are (6.233), (6.234), +(6.235), and(6.236). Wehave found15 solutions wito(6.233) of type A2 +1∗A2in(6.237), +we have found 45 solutions wito (6.233) of type A1∗A3in (6.238), we have found 5 +solutionswito (6.233) of type A2 +2in (6.239), we have found 18 solutions wito (6.233) +of typeA4in (6.240), we have found 5 solutions wito (6.233) of type D4in (6.241), +each giving rise to ( m+ 1)/2 elements in Fix NCm(E8)(ψp). Using the notation from +there, these imply 2 n3,1+n2,2= 2·660+1195 = 2515 solutions for ( wi1,wi2) to (6.234) +withℓT(wi1) +ℓT(wi2) = 4, each giving rise to/parenleftbig(m+1)/2 +2/parenrightbig +elements in Fix NCm(E8)(ψp). +They also imply 3 n2,1,1= 3·2850 = 8550 solutions for ( wi1,wi2,wi3) to (6.235) with +ℓT(wi1)+ℓT(wi2)+ℓT(wi3) = 4, each giving rise to/parenleftbig(m+1)/2 +3/parenrightbig +elements in Fix NCm(E8)(ψp). +Finally, they implyaswell n1,1,1,1= 6750solutionsfor( wi1,wi2,wi3,wi4)to(6.236), each +giving rise to/parenleftbig(m+1)/2 +4/parenrightbig +elements in Fix NCm(E8)(ψp). +In total, we obtain +(15+45+5+18+5)m+1 +2+2515/parenleftbigg(m+1)/2 +2/parenrightbigg ++8550/parenleftbigg(m+1)/2 +3/parenrightbigg ++6750/parenleftbigg(m+1)/2 +4/parenrightbigg +=(m+1)(5m+3)(15m+7)(15m+1) +64 +elements in Fix NCm(E8)(ψp), which agrees with the limit in (9.18h). +Acknowledgements +The authors thank an anonymous referee for a very careful rea ding of the paper [21], +and for the many pertinent suggestions which helped to improve tha t paper and also +this manuscript considerably. +References +[1] G. E. Andrews, The Theory of Partitions , Encyclopedia of Math. and its Applications, vol. 2, +Addison–Wesley, Reading, 1976. +[2] D. Armstrong, Generalized noncrossing partitions and combinatorics of C oxeter groups , Mem. +Amer. Math. Soc., vol. 202, no. 949, Amer. Math. Soc., Providence , R.I., 2009. +[3] D.Armstrong, C.Stump andH. Thomas, A uniform bijection between nonnesting and noncrossing +partitions , Trans. Amer. Math. Soc. (to appear). +[4] C. A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes , +Bull. London Math. Soc. 36(2004), 294–302. +[5] C. A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl gr oups, Trans. +Amer. Math. Soc. 357(2005), 179–196. +[6] D. Bessis, The dual braid monoid , Ann. Sci. ´Ecole Norm. Sup. (4) 36(2003), 647–683. +[7] D. Bessis, Finite complex reflection groups are K(π,1), preprint, arχiv:math/0610777 . +[8] D. Bessis and R. Corran, Non-crossing partitions of type (e,e,r), Adv. Math. 202(2006), 1–49. +[9] D. Bessis and V. Reiner, Cyclic sieving and noncrossing partitions for complex refle ction groups , +Ann. Comb. 15(2011), 197–222. +[10] T. Brady and C. Watt, K(π,1)’s for Artin groups of finite type , Geom. Dedicata 94(2002), +225–250. +[11] F.Chapoton, Enumerative properties of generalized associahedra , S´ eminaireLotharingienCombin. +51(2004), Article B51b, 16 pp. +[12] P. Edelman, Chain enumeration and noncrossing partitions , Discrete Math. 31(1981), 171–180. +[13] S. Fomin and N. Reading, Generalized cluster complexes and Coxeter combinatorics , Int. Math. +Res. Notices 44(2005), 2709–2757.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15 7 +[14] M. Geck, G. Hiss, F. L¨ ubeck, G. Malle and G. Pfeiffer, CHEVIE— a system for computing and +processing generic character tables for finite groups of Lie type, Appl. Algebra Engrg. Comm. +Comput. 7(1996), 175–210. +[15] I. Gordon and S. Griffeth, Catalan numbers for complex reflection groups , Amer. J. Math. (to +appear). +[16] J. E. Humphreys, Reflection groups and Coxeter groups , Cambridge University Press, Cambridge, +1990. +[17] C. Krattenthaler, TheF-triangle of the generalised cluster complex , in: Topics in Discrete Mathe- +matics, dedicated to Jarik Neˇ setˇ ril on the occasion of his 60th bir thday, M. Klazar, J. Kratochvil, +M. Loebl, J. Matouˇ sek, R. Thomas and P. Valtr (eds.), Springer–V erlag, Berlin, New York, 2006, +pp. 93–126. +[18] C. Krattenthaler, TheM-triangle of generalised non-crossing partitions for the t ypesE7andE8, +S´ eminaire Lotharingien Combin. 54(2006), Article B54l, 34 pages. +[19] C. Krattenthaler, Non-crossing partitions on an annulus , in preparation. +[20] C. Krattenthaler and T. W. M¨ uller, Decomposition numbers for finite Coxeter groups and gener- +alised non-crossing partitions , Trans. Amer. Math. Soc. 362(2010), 2723–2787. +[21] C. Krattenthaler and T. W. M¨ uller, Cyclic sieving for generalised non-crossing partitions as soci- +ated with complex reflection groups of exceptional type , in: W80, volume in memory of Herb Wilf, +I. Kotsireas, E. Zima (eds.), Springer–Verlag (to appear); arχiv:1001.0028 . +[22] G. Kreweras, Sur les partitions non crois´ ees d’un cycle , Discrete Math. 1(1972), 333–350. +[23] G. I. Lehrer and J. Michel, Invariant theory and eigenspaces for unitary reflection gro ups, C. R. +Math. Acad. Sci. Paris 336(2003), 795–800. +[24] G. I. Lehrer and D. E. Taylor, Unitary reflection groups , Cambridge University Press, Cambridge, +2009. +[25] N. A. Loehr, Conjectured statistics for the higher q,t-Catalan sequences , Electron. J. Combin. 12 +(2005), Art. #R9, 54 pp. +[26] G. Malle and J. Michel, Constructing representations of Hecke algebras for comple x reflection +groups, LMS J. Comput. Math. 13(2010), 426–450. +[27] I. Marin, The cubic Hecke algebra on at most 5 strands , preprint, arχiv:1110.6621 . +[28] J. Michel, TheGAP-part of the CHEVIEsystem,GAP3-package available for download from +http://people.math.jussieu.fr/jmichel/chevie/chevie .html. +[29] P. Orlik and L. Solomon, Unitary reflection groups and cohomology , Invent. Math. 59(1980), +77–94. +[30] V. Reiner, D. Stanton and D. White, The cyclic sieving phenomenon , J. Combin. Theory Ser. A +108(2004), 17–50. +[31] V. Ripoll, Orbites d’Hurwitz des factorisations primitives d’un ´ el´ ement de Coxeter , J. Algebra +323(2010), 1432–1453. +[32] G. C. Shephard and J. A. Todd, Finite unitary reflection groups , Canad. J. Math. 6(1954), +274–304. +[33] T. A. Springer, Regular elements of finite reflection groups , Invent. Math. 25(1974), 159–198. +[34] J. R. Stembridge, Some hidden relations involving the ten symmetry classes of plane partitions , +J. Combin. Theory Ser. A 68(1994), 372–409. +[35] J.R. Stembridge, Canonical bases and self-evacuating tableaux , DukeMath. J. 82(1996).585–606, +[36] J. R. Stembridge, coxeter,Maplepackagefor workingwith root systems and finite Coxetergroups; +available at http://www.math.lsa.umich.edu/~jrs . +Fakult¨at f¨ur Mathematik, Universit ¨at Wien, Nordbergstraße 15, A-1090 Vienna, +Austria. WWW: http://www.mat.univie.ac.at/ ~kratt. +School of Mathematical Sciences, Queen Mary & Westfield Col lege, University of +London, Mile End Road, London E1 4NS, United Kingdom. +http://www.maths.qmw.ac.uk/ ~twm/. \ No newline at end of file