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Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions. (English) Zbl 1228.05306
Summary: 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 $$c=\sigma_1\sigma_2\dots\sigma_d$$ 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 [Algorithms and Combinatorics 26, 93–126 (2006; Zbl 1122.05096 ); Sémin. Lothar. Comb. 54, B54l, 34 p. (2005; Zbl 1267.05293)]. The type $$A_n$$ decomposition numbers have been computed by I. P. Goulden and D. M. Jackson [Eur. J. Comb. 13, No.5, 357–365 (1992; Zbl 0804.05023)], albeit using a somewhat different language. We explain how to extract the type $$B_n$$ decomposition numbers from results of M. Bóna et al. [Adv. Appl. Math. 24, No. 1, 22–56 (2000; Zbl 0957.05056)] 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 one to address another conjecture of Armstrong on maximal intervals containing a random multi-chain in the generalised non-crossing partitions poset.

##### MSC:
 05E15 Combinatorial aspects of groups and algebras (MSC2010) 05A05 Permutations, words, matrices 05A10 Factorials, binomial coefficients, combinatorial functions 05A15 Exact enumeration problems, generating functions 05A18 Partitions of sets 06A07 Combinatorics of partially ordered sets 20F55 Reflection and Coxeter groups (group-theoretic aspects) 33C05 Classical hypergeometric functions, $${}_2F_1$$
coxeter
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