# Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions C. Krattenthaler† and T. W. Müller Fakultät für Mathematik, Universität 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/. ###### Abstract. Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_{1},T_{2},\dots,T_{d}$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\sigma_{1}\sigma_{2}\cdots\sigma_{d}$ of a Coxeter element $c$ of $W$, such that $\sigma_{i}$ is a Coxeter element in a subgroup of type $T_{i}$ in $W$, $i=1,2,\dots,d$, and such that the factorisation is “minimal” in the sense that the sum of the ranks of the $T_{i}$’s, $i=1,2,\dots,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author in [*“Topics in Discrete Mathematics,”* M. Klazar et al. (eds.), Springer–Verlag, Berlin, New York, 2006, pp. 93–126] and [*Séminaire Lotharingien Combin.* **54** (2006), Article B54l]. The type $A_{n}$ decomposition numbers have been computed by Goulden and Jackson in [*Europ. J. Combin.* **13** (1992), 357–365], albeit using a somewhat different language. We explain how to extract the type $B_{n}$ decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. **24** (2000), 22–56] on map enumeration. Our formula for the type $D_{n}$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_{1}\leq r_{2}\leq\dots\leq r_{l}$, the number of multi-chains $\pi_{1}\leq\pi_{2}\leq\dots\leq\pi_{l}$ in Armstrong’s generalised non-crossing partitions poset, where the poset rank of $\pi_{i}$ equals $r_{i}$, and where the “block structure” of $\pi_{1}$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_{n}$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong’s $F=M$ Conjecture in type $D_{n}$, thus completing a computational proof of the $F=M$ Conjecture for all types. It also allows to address another conjecture of Armstrong on maximal intervals containing a random multichain in the generalised non-crossing partitions poset. ###### Key words and phrases: root systems, reflection groups, Coxeter groups, generalised non-crossing partitions, annular non-crossing partitions, chain enumeration, Möbius function, $M$-triangle, generalised cluster complex, face numbers, $F$-triangle, Chu–Vandermonde summation ###### 2000 Mathematics Subject Classification: Primary 05E15; Secondary 05A05 05A10 05A15 05A18 06A07 20F55 33C05 †Research partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory” ## 1. Introduction The introduction of *non-crossing partitions for finite reflection groups* (finite Coxeter groups) by Bessis [[8](#bib.bib8)] and Brady and Watt [[15](#bib.bib15)] marks the creation of a new, exciting subject of combinatorial theory, namely the study of these new combinatorial objects which possess numerous beautiful properties, and seem to relate to several other objects of combinatorics and algebra, most notably to the cluster complex of Fomin and Zelevinsky [[21](#bib.bib21)] (cf. [[2](#bib.bib2), [3](#bib.bib3), [5](#bib.bib5), [4](#bib.bib4), [8](#bib.bib8), [9](#bib.bib9), [14](#bib.bib14), [15](#bib.bib15), [16](#bib.bib16), [17](#bib.bib17), [20](#bib.bib20)]). They reduce to the classical non-crossing partitions of Kreweras [[30](#bib.bib30)] for the irreducible reflection groups of type $A_{n}$ (i.e., the symmetric groups), and to Reiner’s [[32](#bib.bib32)] type $B_{n}$ non-crossing partitions for the irreducible reflections groups of type $B_{n}$. (They differ, however, from the type $D_{n}$ non-crossing partitions of [[32](#bib.bib32)].) The subject has been enriched by Armstrong through introduction of his *generalised non-crossing partitions for reflection groups* in [[1](#bib.bib1)]. In the symmetric group case, these reduce to the $m$-divisible non-crossing partitions of Edelman [[18](#bib.bib18)], while they produce new combinatorial objects already for the reflection groups of type $B_{n}$. Again, these generalised non-crossing partitions possess numerous beautiful properties, and seem to relate to several other objects of combinatorics and algebra, most notably to the generalised cluster complex of Fomin and Reading [[19](#bib.bib19)] (cf. [[1](#bib.bib1), [6](#bib.bib6), [7](#bib.bib7), [20](#bib.bib20), [27](#bib.bib27), [28](#bib.bib28), [29](#bib.bib29), [36](#bib.bib36), [37](#bib.bib37), [38](#bib.bib38)]). From a technical point of view, the *main subject matter* of the present paper is the computation of the number of certain factorisations of the Coxeter element of a reflection group. These *decomposition numbers*, as we shall call them from now on (see Section [2](#S2) for the precise definition), arose in [[27](#bib.bib27), [28](#bib.bib28)], where it was shown that they play a crucial role in the computation of enumerative invariants of (generalised) non-crossing partitions. Moreover, in these two papers the decomposition numbers for the exceptional reflection groups have been computed, and it was pointed out that the decomposition numbers in type $A_{n}$ (i.e., the decomposition numbers for the symmetric groups) had been earlier computed by Goulden and Jackson in [[23](#bib.bib23)]. Here, we explain how the decomposition numbers in type $B_{n}$ can be extracted from results of Bóna, Bousquet, Labelle and Leroux [[12](#bib.bib12)] on the enumeration of certain planar maps, and we find formulae for the decomposition numbers in type $D_{n}$, thus completing the project of computing the decomposition numbers for all the irreducible reflection groups. The *main goal* of the present paper, however, is to access the enumerative theory of the generalised non-crossing partitions of Armstrong via these decomposition numbers. Indeed, one finds numerous enumerative results on ordinary and generalised non-crossing partitions in the literature (cf. [[1](#bib.bib1), [2](#bib.bib2), [4](#bib.bib4), [8](#bib.bib8), [9](#bib.bib9), [18](#bib.bib18), [30](#bib.bib30), [32](#bib.bib32), [37](#bib.bib37)]): results on the total number of (generalised) non-crossing partitions of a given size, of those with a fixed number of blocks, of those with a given block structure, results on the number of (multi-)chains of a given length in a given poset of (generalised) non-crossing partitions, results on rank-selected chain enumeration (that is, results on the number of chains in which the ranks of the elements of the chains have been fixed), etc. We show that not only can *all* these results be rederived from our decomposition numbers, we are also able to find several new enumerative results. In this regard, the most general type of result that we find is formulae for the number of (multi-)chains $\pi_{1}\leq\pi_{2}\leq\dots\leq\pi_{l-1}$ in the poset of non-crossing partitions of type $A_{n}$, $B_{n}$, respectively $D_{n}$, in which the block structure of $\pi_{1}$ is fixed as well as the ranks of $\pi_{2},\dots,\pi_{l-1}$. Even the corresponding result in type $A_{n}$, for the non-crossing partitions of Kreweras, is new. Furthermore, from the result in type $D_{n}$, by a suitable summation, we are able to find a formula for the rank-selected chain enumeration in the poset of generalised non-crossing partitions of type $D_{n}$, thus generalising the earlier formula of Athanasiadis and Reiner [[4](#bib.bib4)] for the rank-selected chain enumeration of “ordinary” non-crossing partitions of type $D_{n}$. In conjunction with the results from [[27](#bib.bib27), [28](#bib.bib28)], this generalisation in turn allows us to complete a computational case-by-case proof of Armstrong’s “$F=M$ Conjecture” [[1](#bib.bib1), Conjecture 5.3.2] predicting a surprising relationship between a certain face count in the generalised cluster complex of Fomin and Reading and the Möbius function in the poset of generalised non-crossing partitions of Armstrong. (A case-free proof had been found earlier by Tzanaki in [[38](#bib.bib38)].) Our results allow us also to address another conjecture of Armstrong [[1](#bib.bib1), Conj. 3.5.13] on maximal intervals containing a random multichain in the poset of generalised non-crossing partitions. We show that the conjecture is indeed true for types $A_{n}$ and $B_{n}$, but that it fails for type $D_{n}$ (and we suspect that it will also fail for most of the exceptional types). We remark that a totally different approach to the enumerative theory of (generalised) non-crossing partitions is proposed in [[29](#bib.bib29)]. This approach is, however, completely combinatorial and avoids, in particular, reflection groups. It is, therefore, not capable of computing our decomposition numbers nor anything else which is intrinsic to the combinatorics of reflection groups. A similar remark applies to [[31](#bib.bib31), Theorem 4.1], where a remarkable uniform recurrence is found for rank-selected chain enumeration in the generalised non-crossing partitions of *any* type. It could be used, for example, for verifying our result in Corollary [19](#Thmtheorem19) on the rank-selected chain enumeration in the generalised non-crossing partitions of type $D_{n}$, but it is not capable of computing our decomposition numbers nor of verifying results with restrictions on block structure. Our paper is organised as follows. In the next section we define the decomposition numbers for finite reflection groups from [[27](#bib.bib27), [28](#bib.bib28)], the central objects in our paper, together with a combinatorial variant, which depends on combinatorial realisations of non-crossing partitions, which we also explain in the same section. This is followed by an intermediate section in which we collect together some auxiliary results that will be needed later on. In Section [4](#S4), we recall Goulden and Jackson’s formula [[23](#bib.bib23)] for the full rank decomposition numbers of type $A_{n}$, together with the formula from [[28](#bib.bib28), Theorem 10] that it implies for the decomposition numbers of type $A_{n}$ of arbitrary rank. The purpose of Section [5](#S5) is to explain how formulae for the decomposition numbers of type $B_{n}$ can be extracted from results of Bóna, Bousquet, Labelle and Leroux in [[12](#bib.bib12)]. The type $D_{n}$ decomposition numbers are computed in Section [6](#S6). The approach that we follow is, essentially, the approach of Goulden and Jackson in [[23](#bib.bib23)]: we translate the counting problem into the problem of enumerating certain maps. This problem is then solved by a combinatorial decomposition of these maps, translating the decomposition into a system of equations for corresponding generating functions, and finally solving this system with the help of the multidimensional Lagrange inversion formula of Good. Sections [7](#S7)–[11](#S11) form the “applications part” of the paper. In the preparatory Section [7](#S7), we recall the definition of the generalised non-crossing partitions of Armstrong, and we explain the combinatorial realisations of the generalised non-crossing partitions for the types $A_{n}$, $B_{n}$, and $D_{n}$ from [[1](#bib.bib1)] and [[29](#bib.bib29)]. The bulk of the applications is contained in Section [8](#S8), where we present three theorems, Theorems [11](#Thmtheorem11), [13](#Thmtheorem13), and [15](#Thmtheorem15), on the number of factorisations of a Coxeter element of type $A_{n}$, $B_{n}$, respectively $D_{n}$, with less stringent restrictions on the factors than for the decomposition numbers. These theorems result from our formulae for the (combinatorial) decomposition numbers upon appropriate summations. Subsequently, it is shown that the corresponding formulae imply all known enumeration results on non-crossing partitions and generalised non-crossing partitions, plus several new ones, see Corollaries [12](#Thmtheorem12), [14](#Thmtheorem14), [16](#Thmtheorem16)–[19](#Thmtheorem19) and the accompanying remarks. Section [9](#S9) presents the announced computational proof of the $F=M$ (ex-)Conjecture for type $D_{n}$, based on our formula in Corollary [19](#Thmtheorem19) for the rank-selected chain enumeration in the poset of generalised non-crossing partitions of type $D_{n}$, while Section [10](#S10) addresses Conjecture 3.5.13 from [[1](#bib.bib1)], showing that it does not hold in general since it fails in type $D_{n}$. In the final Section [11](#S11) we point out that the decomposition numbers do not only allow one to derive enumerative results for the generalised non-crossing partitions of the classical types, they also provide all the means for doing this for the exceptional types. For the convenience of the reader, we list the values of the decomposition numbers for the exceptional types that have been computed in [[27](#bib.bib27), [28](#bib.bib28)] in an appendix. In concluding the introduction, we want to attract the reader’s attention to the fact that many of the formulae presented here are very combinatorial in nature (see Sections [4](#S4), [5](#S5), [8](#S8)). This raises the natural question as to whether it is possible to find combinatorial proofs for them. Indeed, a combinatorial (and, in fact, almost bijective) proof of the formula of Goulden and Jackson, presented here in Theorem [5](#Thmtheorem5), has been given by Bousquet, Chauve and Schaeffer in [[13](#bib.bib13)]. Moreover, most of the proofs for the known enumeration results on (generalised) non-crossing partitions presented in [[1](#bib.bib1), [2](#bib.bib2), [4](#bib.bib4), [18](#bib.bib18), [32](#bib.bib32)] are combinatorial. On the other hand, to our knowledge so far nobody has given a combinatorial proof for Theorem [7](#Thmtheorem7), the formula for the decomposition numbers of type $B_{n}$, essentially due to Bóna, Bousquet, Labelle and Leroux [[12](#bib.bib12)], although we believe that this should be possible by modifying the ideas from [[13](#bib.bib13)]. There are also other formulae in our paper (see e.g. Corollaries [12](#Thmtheorem12) and [14](#Thmtheorem14), Eqs. ([6.1](#S6.E1)) and ([8.33](#S8.E33))) which seem amenable to combinatorial proofs. However, to find combinatorial proofs for our type $D_{n}$ results (cf. in particular Theorem [9](#Thmtheorem9).(ii) and Corollaries [16](#Thmtheorem16)–[19](#Thmtheorem19)) seems rather hopeless to us. ## 2. Decomposition numbers for finite Coxeter groups In this section, we introduce the decomposition numbers from [[27](#bib.bib27), [28](#bib.bib28)], which are (Coxeter) group-theoretical in nature, plus combinatorial variants for Coxeter groups of types $B_{n}$ and $D_{n}$, which will be important in combinatorial applications. These variants depend on the combinatorial realisation of these Coxeter groups, which we also explain here. Let $\Phi$ be a finite root system of rank $n$. (We refer the reader to [[24](#bib.bib24)] for all terminology on root systems.) For an element $\alpha\in\Phi$, let $t_{\alpha}$ denote the corresponding reflection in the central hyperplane perpendicular to $\alpha$. Let $W=W(\Phi)$ be the group generated by these reflections. As is well-known (cf. e.g. [[24](#bib.bib24), Sec. 6.4]), any such reflection group is at the same time a finite Coxeter group, and all finite Coxeter groups can be realised in this way. By definition, any element $w$ of $W$ can be represented as a product $w=t_{1}t_{2}\cdots t_{\ell}$, where the $t_{i}$’s are reflections. We call the minimal number of reflections which is needed for such a product representation the *absolute length* of $w$, and we denote it by $\ell_{T}(w)$. We then define the *absolute order* on $W$, denoted by $\leq_{T}$, via [TABLE] As is well-known and easy to see, this is equivalent to the statement that every shortest representation of $u$ by reflections occurs as an initial segment in some shortest product representation of $w$ by reflections. Now, for a finite root system $\Phi$ of rank $n$, types $T_{1},T_{2},\dots,T_{d}$ (in the sense of the classification of finite Coxeter groups), and a *Coxeter element* $c$, the *decomposition number* $N_{\Phi}(T_{1},T_{2},\dots,T_{d})$ is defined as the number of “minimal” products $c_{1}c_{2}\cdots c_{d}$ less than or equal to $c$ in absolute order, “minimal” meaning that $\ell_{T}(c_{1})+\ell_{T}(c_{2})+\dots+\ell_{T}(c_{d})=\ell_{T}(c_{1}c_{2}\cdots c_{d})$, such that, for $i=1,2,\dots,d$, the type of $c_{i}$ as a parabolic Coxeter element is $T_{i}$. (Here, the term “parabolic Coxeter element” means a Coxeter element in some parabolic subgroup. The reader should recall that it follows from [[8](#bib.bib8), Lemma 1.4.3] that any element $c_{i}$ is indeed a Coxeter element in a parabolic subgroup of $W=W(\Phi)$. By definition, the type of $c_{i}$ is the type of this parabolic subgroup. The reader should also note that, because of the rewriting [TABLE] any $c_{i}$ in such a minimal product $c_{1}c_{2}\cdots c_{d}\leq_{T}c$ is itself $\leq_{T}c$.) It is easy to see that the decomposition numbers are independent of the choice of the Coxeter element $c$. (This follows from the well-known fact that any two Coxeter elements are conjugate to each other; cf. [[24](#bib.bib24), Sec. 3.16].) The decomposition numbers satisfy several linear relations between themselves. First of all, the number $N_{\Phi}(T_{1},T_{2},\dots,T_{d})$ is independent of the order of the types $T_{1},T_{2},\dots,\break T_{d}$; that is, we have [TABLE] for every permutation $\sigma$ of $\{1,2,\dots,d\}$. This is, in fact, a consequence of the rewriting ([2.1](#S2.E1)). Furthermore, by the definition of these numbers, those of “lower rank” can be computed from those of “full rank.” To be precise, we have [TABLE] where the sum is taken over all types $T$ of rank $n-\operatorname{rk}T_{1}-\operatorname{rk}T_{2}-\dots-\operatorname{rk}T_{d}$ (with $\operatorname{rk}T$ denoting the rank of the root system $\Psi$ of type $T$, and $n$ still denoting the rank of the fixed root system $\Phi$; for later use we record that [TABLE] for any parabolic Coxeter element $w_{0}$ of type $T_{0}$). The decomposition numbers for the exceptional types have been computed in [[27](#bib.bib27), [28](#bib.bib28)]. For the benefit of the reader, we reproduce these numbers in the appendix. The decomposition numbers for type $A_{n}$ are given in Section [4](#S4), the ones for type $B_{n}$ are computed in Section [5](#S5), while the ones for type $D_{n}$ are computed in Section [6](#S6). Next we introduce variants of the above decomposition numbers for the types $B_{n}$ and $D_{n}$, which depend on the combinatorial realisation of the Coxeter groups of these types. As is well-known, the reflection group $W(A_{n})$ can be realised as the symmetric group $S_{n+1}$ on $\{1,2,\dots,n+1\}$. The reflection groups $W(B_{n})$ and $W(D_{n})$, on the other hand, can be realised as subgroups of the symmetric group on $2n$ elements. (See e.g. [[11](#bib.bib11), Sections 8.1 and 8.2].) Namely, the reflection group $W(B_{n})$ can be realised as the subgroup of the group of all permutations $\pi$ of [TABLE] satisfying the property [TABLE] (Here, and in what follows, $\bar{\bar{i}}$ is identified with $i$ for all $i$.) In this realisation, there is an analogue of the disjoint cycle decomposition of permutations. Namely, every $\pi\in W(B_{n})$ can be decomposed as [TABLE] where, for $i=1,2,\dots,s$, $\kappa_{i}$ is of one of two possible types of “cycles”: a *type $A$ cycle*, by which we mean a permutation of the form [TABLE] or a *type $B$ cycle*, by which we mean a permutation of the form [TABLE] $a_{1},a_{2},\dots,a_{k}\in\{1,2,\dots,n,\bar{1},\bar{2},\dots,\bar{n}\}$. (Here we adopt notation from [[15](#bib.bib15)].) In both cases, we call $k$ the *length* of the “cycle.” The decomposition ([2.6](#S2.E6)) is unique up to a reordering of the $\kappa_{i}$’s. We call a type $A$ cycle of length $k$ *of combinatorial type* $A_{k-1}$, while we call a type $B$ cycle of length $k$ *of combinatorial type* $B_{k}$, $k=1,2,\dots$. The reader should observe that, when regarded as a parabolic Coxeter element, for $k\geq 2$ a type $A$ cycle of length $k$ has type $A_{k-1}$, while a type $B$ cycle of length $k$ has type $B_{k}$. However, a type $B$ cycle of length $1$, that is, a permutation of the form $(i,\bar{i})$, has type $A_{1}$ when regarded as a parabolic Coxeter element, while we say that it has combinatorial type $B_{1}$. (The reader should recall that, in the classification of finite Coxeter groups, the type $B_{1}$ does not occur, respectively, that sometimes $B_{1}$ is identified with $A_{1}$. Here, when we speak of “combinatorial type,” then we *do* distinguish between $A_{1}$ and $B_{1}$. For example, the “cycles” $((1,2))=(1,2)\,(\bar{1},\bar{2})$ or $((\bar{1},2))=(\bar{1},2)\,(1,\bar{2})$ have combinatorial type $A_{1}$, whereas the cycles $[1]=(1,\bar{1})$ or $[2]=(2,\bar{2})$ have combinatorial type $B_{1}$.) As Coxeter element for $W(B_{n})$, we choose [TABLE] Now, given *combinatorial* types $T_{1},T_{2},\dots,T_{d}$, each of which being a product of $A_{k}$’s and $B_{k}$’s, $k=1,2,\dots$, the *combinatorial decomposition number* $N_{B_{n}}^{\text{comb}}(T_{1},T_{2},\dots,T_{d})$ is defined as the number of minimal products $c_{1}c_{2}\cdots c_{d}$ less than or equal to $c$ in absolute order, where “minimal” has the same meaning as above, such that for $i=1,2,\dots,d$ the *combinatorial* type of $c_{i}$ is $T_{i}$. Because of ([2.1](#S2.E1)), the combinatorial decomposition numbers $N_{B_{n}}^{\text{comb}}(T_{1},T_{2},\dots,T_{d})$ satisfy also ([2.2](#S2.E2)) and ([2.3](#S2.E3)). The reflection group $W(D_{n})$ can be realised as the subgroup of the group of all permutations $\pi$ of $\{1,2,\dots,n,\bar{1},\bar{2},\dots,\bar{n}\}$ satisfying ([2.5](#S2.E5)) and the property that an even number of elements from $\{1,2,\dots,n\}$ is mapped to an element of negative sign. (Here, the elements $1,2,\dots,n$ are considered to have sign $+$, while the elements $\bar{1},\bar{2},\dots,\bar{n}$ are considered to have sign $-$.) Since $W(D_{n})$ is a subgroup of $W(B_{n})$, and since the above realisation of $W(D_{n})$ is contained as a subset in the realisation of $W(B_{n})$ that we just described, any $\pi\in W(D_{n})$ can be decomposed as in ([2.6](#S2.E6)), where, for $i=1,2,\dots,d$, $\kappa_{i}$ is either a type $A$ or a type $B$ cycle. Requiring that $\pi$ is in the subgroup $W(D_{n})$ of $W(B_{n})$ is equivalent to requiring that there is an even number of type $B$ cycles in the decomposition ([2.6](#S2.E6)). Again, the decomposition ([2.6](#S2.E6)) for $\pi\in W(D_{n})$ is unique up to a reordering of the $\kappa_{i}$’s. As Coxeter element, we choose [TABLE] We shall be entirely concerned with elements $\pi$ of $W(D_{n})$ which are less than or equal to $c$. It is not difficult to see (and it is shown in [[4](#bib.bib4), Sec. 3]) that the unique factorisation of any such element $\pi$ has either [math] or $2$ type $B$ cycles, and in the latter case one of the type $B$ cycles is $[n]=(n,\bar{n})$. In this latter case, in abuse of terminology, we call the product of these two type $B$ cycles, $[a_{1},a_{2},\dots,a_{k-1}]\,[n]$ say, a “cycle” *of combinatorial type* $D_{k}$. More generally, we shall say for any product of two disjoint type $B$ cycles of the form [TABLE] that it is a “cycle” of *combinatorial type* $D_{k}$. The reader should observe that, when regarded as parabolic Coxeter element, for $k\geq 4$ an element of the form ([2.9](#S2.E9)) has type $D_{k}$. However, if $k=3$, it has type $A_{3}$ when regarded as parabolic Coxeter element, while we say that it has combinatorial type $D_{3}$, and, if $k=2$, it has type $A_{1}^{2}$ when regarded as parabolic Coxeter element, while we say that it has combinatorial type $D_{2}$. (The reader should recall that, in the classification of finite Coxeter groups, the types $D_{3}$ and $D_{2}$ do not occur, respectively, that sometimes $D_{3}$ is identified with $A_{3}$, $D_{2}$ being identified with $A_{1}^{2}$. Here, when we speak of “combinatorial type,” then we *do* distinguish between $D_{3}$ and $A_{3}$, and between $D_{2}$ and $A_{1}^{2}$.) Now, given *combinatorial* types $T_{1},T_{2},\dots,T_{d}$, each of which being a product of $A_{k}$’s and $D_{k}$’s, $k=1,2,\dots$, the *combinatorial decomposition number* $N_{B_{n}}^{\text{comb}}(T_{1},T_{2},\dots,T_{d})$ is defined as the number of minimal products $c_{1}c_{2}\cdots c_{d}$ less than or equal to $c$ in absolute order, where “minimal” has the same meaning as above, such that for $i=1,2,\dots,d$ the *combinatorial* type of $c_{i}$ is $T_{i}$. Because of ([2.1](#S2.E1)), the combinatorial decomposition numbers $N_{D_{n}}^{\text{comb}}(T_{1},T_{2},\dots,T_{d})$ satisfy also ([2.2](#S2.E2)) and ([2.3](#S2.E3)). ## 3. Auxiliary results In our computations in the proof of Theorem [9](#Thmtheorem9), leading to the determination of the decomposition numbers of type $D_{n}$, we need to apply the Lagrange–Good inversion formula [[22](#bib.bib22)] (see also [[26](#bib.bib26), Sec. 5] and the references cited therein). We recall it here for the convenience of the reader. In doing so, we use standard multi-index notation. Namely, given a positive integer $d$, and vectors $\mathbf{z}=(z_{1},z_{2},\dots,z_{d})$ and $\mathbf{n}=(n_{1},n_{2},\dots,n_{d})$, we write $\mathbf{z}^{\mathbf{n}}$ for $z_{1}^{n_{1}}z_{2}^{n_{2}}\cdots z_{d}^{n_{d}}$. Furthermore, in abuse of notation, given a formal power series $f$ in $d$ variables, $f(\mathbf{z})$ stands for $f(z_{1},z_{2},\dots,z_{d})$. Moreover, given $d$ formal power series $f_{1},f_{2},\dots,f_{d}$ in $d$ variables, $\mathbf{f}^{\mathbf{n}}(\mathbf{z})$ is short for [TABLE] Finally, if $\mathbf{m}=(m_{1},m_{2},\dots,m_{d})$ is another vector, then $\mathbf{m}+\mathbf{n}$ is short for $(m_{1}+n_{1},m_{2}+n_{2},\dots,m_{d}+n_{d})$. Notation such as $\mathbf{m}-\mathbf{n}$ has to be interpreted in a similar way. ###### **Theorem 1**** **(Lagrange–Good inversion)**.** *Let $d$ be a positive integer, and let $f_{1}(\mathbf{z}),f_{2}(\mathbf{z}),\dots,f_{d}(\mathbf{z})$ be formal power series in $\mathbf{z}=(z_{1},z_{2},\dots,z_{d})$ with the property that, for all $i$, $f_{i}(\mathbf{z})$ is of the form $z_{i}/\varphi_{i}(\mathbf{z})$ for some formal power series $\varphi_{i}(\mathbf{z})$ with $\varphi_{i}(0,0,\dots,0)\neq 0$. Then, if we expand a formal power series $g(\mathbf{z})$ in terms of powers of the $f_{i}(\mathbf{z}),$* [TABLE] *the coefficients $\gamma_{\mathbf{n}}$ are given by* [TABLE] *where $\mathbf{e}=(1,1,\dots,1),$ where the sum in ([3.1](#S3.E1)) runs over all $d$-tuples $\mathbf{n}$ of non-negative integers, and where $\left\langle\mathbf{z}^{\mathbf{m}}\right\rangle h(\mathbf{z})$ denotes the coefficient of $\mathbf{z}^{\mathbf{m}}$ in the formal Laurent series $h(\mathbf{z})$.* Next, we prove a determinant lemma and a corollary, both of which will also be used in the proof of Theorem [9](#Thmtheorem9). ###### **Lemma 2****.** *Let $d$ be a positive integer, and let $X_{1},X_{2},\dots,X_{d},Y_{2},Y_{3},\dots,Y_{d}$ be indeterminates. Then* [TABLE] *where $\chi(\mathcal{S})=1$ if $\mathcal{S}$ is true and $\chi(\mathcal{S})=0$ otherwise.* ###### Proof. By using multilinearity in the rows, we rewrite the determinant on the left-hand side of ([3.2](#S3.E2)) as [TABLE] Next, we subtract the first column from all other columns. As a result, we obtain the determinant [TABLE] Now we add rows $2,3,\dots,d$ to the first row. After that, our determinant becomes lower triangular, with the entry in the first row and column equal to $\sum_{i=1}^{d}X_{i}-\sum_{i=2}^{d}Y_{i}$, and the diagonal entry in row $i$, $i\geq 2$, equal to $Y_{i}$. Hence, we obtain the claimed result. ∎ ###### **Corollary 3****.** *Let $d$ and $r$ be positive integers, $1\leq r\leq d,$ and let $X_{1},X_{2},\dots,X_{d},$ $Y$ and $Z$ be indeterminates. Then, with notation as in Lemma [2](#Thmtheorem2),* [TABLE] ###### Proof. We write the diagonal entry in the $r$-th row of the determinant in ([3.3](#S3.E3)) as [TABLE] and then use linearity of the determinant in the $r$-th row to decompose the determinant as [TABLE] where $D_{1}$ is the determinant in ([3.2](#S3.E2)) with $X_{r}$ replaced by $X_{r}+Y-Z$, and with $Y_{i}=Y$ for all $i$, and where $D_{2}$ is the determinant in ([3.2](#S3.E2)) with $d$ replaced by $d-1$, with $Y_{i}=Y$ for all $i$, and with $X_{i}$ replaced by $X_{i-1}$ for $i=r+1,r+2,\dots,d$. Hence, using Lemma [2](#Thmtheorem2), we deduce that the determinant in ([3.3](#S3.E3)) is equal to [TABLE] Little simplification then leads to ([3.3](#S3.E3)). ∎ We end this section with a summation lemma, which we shall need in Sections [5](#S5) and [6](#S6) in order to compute the $B_{n}$, respectively $D_{n}$, decomposition numbers of arbitrary rank from those of full rank, and in Section [8](#S8) to derive enumerative results for (generalised) non-crossing partitions from our formulae for the decomposition numbers. ###### **Lemma 4****.** *Let $M$ and $r$ be non-negative integers. Then* [TABLE] *where the multinomial coefficient is defined by* [TABLE] ###### Proof. The identity results directly by comparing coefficients of $z^{r}$ on both sides of the identity [TABLE] ∎ ## 4. Decomposition numbers for type $A$ As was already pointed out in [[28](#bib.bib28), Sec. 10], the decomposition numbers for type $A_{n}$ have already been computed by Goulden and Jackson in [[23](#bib.bib23), Theorem 3.2], albeit using a somewhat different language. (The condition on the sum $l(\alpha_{1})+l(\alpha_{2})+\dots+l(\alpha_{m})$ is misstated throughout the latter paper. It should be replaced by $l(\alpha_{1})+l(\alpha_{2})+\dots+l(\alpha_{m})=(m-1)n+1$.) In our terminology, their result reads as follows. ###### **Theorem 5****.** *Let $T_{1},T_{2},\dots,T_{d}$ be types with $\operatorname{rk}T_{1}+\operatorname{rk}T_{2}+\dots+\operatorname{rk}T_{d}=n,$ where* [TABLE] *Then* [TABLE] *where the multinomial coefficient is defined as in Lemma [4](#Thmtheorem4).* Here we have used Stembridge’s [[35](#bib.bib35)] notation for the decomposition of types into a product of irreducibles; for example, the equation $T=A_{2}^{3}*A_{5}$ means that the root system of type $T$ decomposes into the orthogonal product of $3$ copies of root systems of type $A_{2}$ and one copy of the root system of type $A_{5}$. It was shown in [[28](#bib.bib28), Theorem 10] that, upon applying the summation formula in Lemma [4](#Thmtheorem4) to the result in Theorem [5](#Thmtheorem5) in a suitable manner, one obtains a compact formula for *all* type $A_{n}$ decomposition numbers. ###### **Theorem 6****.** *Let the types $T_{1},T_{2},\dots,T_{d}$ be given, where* [TABLE] *Then* [TABLE] *where the multinomial coefficient is defined as in Lemma [4](#Thmtheorem4). All other decomposition numbers $N_{A_{n}}(T_{1},T_{2},\dots,T_{d})$ are zero.* ## 5. Decomposition numbers for type $B$ In this section we compute the decomposition numbers in type $B_{n}$. We show that one can extract the corresponding formulae from results of Bóna, Bousquet, Labelle and Leroux [[12](#bib.bib12)] on the enumeration of certain planar maps, which they call $m$-ary cacti. While reading the statement of the theorem, the reader should recall from Section [2](#S2) the distinction between group-theoretic and combinatorial decomposition numbers. ###### **Theorem 7****.** (i)* If $T_{1},T_{2},\dots,T_{d}$ are types with $\operatorname{rk}T_{1}+\operatorname{rk}T_{2}+\dots+\operatorname{rk}T_{d}=n,$ where* [TABLE] *and* [TABLE] *for some $\alpha\geq 1,$ then* [TABLE] *where the multinomial coefficient is defined as in Lemma [4](#Thmtheorem4). For $\alpha\geq 2,$ the number $N_{B_{n}}(T_{1},T_{2},\dots,T_{d})$ is given by the same formula.* (ii)* If $T_{1},T_{2},\dots,T_{d}$ are types with $\operatorname{rk}T_{1}+\operatorname{rk}T_{2}+\dots+\operatorname{rk}T_{d}=n,$ where* [TABLE] *then* [TABLE] *where $m^{(j)}_{0}=n-\operatorname{rk}T_{j}-\sum_{s=1}^{n}m^{(j)}_{s}$.* (iii)* All other decomposition numbers $N_{B_{n}}(T_{1},T_{2},\dots,T_{d})$ and $N_{B_{n}}^{\text{comb}}(T_{1},T_{2},\dots,T_{d})$ with $\operatorname{rk}T_{1}+\operatorname{rk}T_{2}+\dots+\operatorname{rk}T_{d}=n$ are zero.* ###### Proof. Determining the decomposition numbers [TABLE] (recall ([2.2](#S2.E2))), respectively [TABLE] amounts to counting all possible factorisations [TABLE] where $\sigma_{i}$ has type $T_{i}$ as a parabolic Coxeter element, respectively has combinatorial type $T_{i}$. The reader should observe that the factorisation ([5.3](#S5.E3)) is *minimal*, in the sense that [TABLE] since $\ell_{T}(\sigma_{i})=\operatorname{rk}T_{i}$, and since, by our assumption, the sum of the ranks of the $T_{i}$’s equals $n$. A further observation is that, in a factorisation ([5.3](#S5.E3)), there must be at least one factor $\sigma_{i}$ which contains a type $B$ cycle in its (type $B$) disjoint cycle decomposition, because the sign of $[1,2,\dots,n]$ as an element of the group $S_{2n}$ of all permutations of $\{1,2,\dots,n,\bar{1},\bar{2},\dots,\bar{n}\}$ is $-1$, while the sign of any type $A$ cycle is $+1$. We first prove Claim (iii). Let us assume, by contradiction, that there is a minimal decomposition ([5.3](#S5.E3)) in which, altogether, we find *at least two* type $B$ cycles in the (type $B$) disjoint cycle decompositions of the $\sigma_{i}$’s. In that case, ([5.3](#S5.E3)) has the form [TABLE] where $\kappa_{1}$ and $\kappa_{2}$ are two type $B$ cycles, and $u_{1},u_{2},u_{3}$ are the factors in between. Moreover, the factorisation ([5.4](#S5.E4)) is minimal, meaning that [TABLE] We may rewrite ([5.4](#S5.E4)) as [TABLE] or, setting $u_{1}^{\prime}=\kappa_{2}^{-1}\kappa_{1}^{-1}u_{1}\kappa_{1}\kappa_{2}$ and $u_{2}^{\prime}=\kappa_{2}^{-1}u_{2}\kappa_{2}$, as [TABLE] This factorisation is still minimal since $u_{1}^{\prime}$ is conjugate to $u_{1}$ and $u_{2}^{\prime}$ is conjugate to $u_{2}$. At this point, we observe that $\kappa_{1}$ must be a cycle of the form ([2.8](#S2.E8)) with $a_{1}\{i_{h+1}-1\}_{d}$, the sequence of colours $\{i_{h}+1\}_{d},\dots,\{i_{h+1}-1\}_{d}$ must be interpreted “cyclically,” that is, as $\{i_{h}+1\}_{d},\{i_{h}+1\}_{d}+1,\dots,d,1,2,\dots,\{i_{h+1}-1\}_{d}$. As we observed above, the number of edges bounding the inner face is $2d$. On the other hand, using the notation just introduced, this number also equals [TABLE] Hence, there is precisely one $h$ for which $i_{h+1}1$, there are many different minimal elements. In particular, $NC^{m}(\Phi)$ has no least element if $m>1$; hence, $NC^{m}(\Phi)$ is not a lattice for $m>1$. (It is, however, a graded join-semilattice, see [[1](#bib.bib1), Theorem 3.4.4].) In what follows, we shall use the notions “generalised non-crossing partitions” and “$m$-divisible non-crossing partitions” interchangeably, where the latter notion will be employed particularly in contexts in which we want to underline the presence of the parameter $m$. In the remainder of this section, we explain combinatorial realisations of the $m$-divisible non-crossing partitions of types $A_{n-1}$, $B_{n}$, and $D_{n}$. In order to be able to do so, we need to recall the definition of Kreweras’ non-crossing partitions of $\{1,2,\dots,N\}$, his *“partitions non croisées d’un cycle”* of [[30](#bib.bib30)]. We place $N$ vertices around a cycle, and label them $1,2,\dots,N$ in clockwise order. The circular representation of a partition of the set $\{1,2,\dots,N\}$ is the geometric object which arises by representing each block $\{i_{1},i_{2},\dots,i_{k}\}$ of the partition, where $i_{1}