# Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

### (62 pages)

Abstract. Given a finite irreducible Coxeter group W, a positive integer d, and types T1,T2,...,Td (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c=\si1\si2...\sid of a Coxeter element c of W, such that \sii is a Coxeter element in a subgroup of type Ti in W, i=1,2,...,d, and such that the factorisation is "minimal" in the sense that the sum of the ranks of the Ti's, i=1,2,...,d, equals the rank of W. For the exceptional types, these decomposition numbers had 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]. We explain that the type An decomposition numbers had been computed by Goulden and Jackson in [Europ. J. Combin. 13 (1992), 357-365], and that the type Bn decomposition numbers can be extracted from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. 24 (2000), 22-56] on map enumeration. Our formula for the type Dn decomposition numbers is new. These results are then used to determine, for a fixed positive integer l and fixed integers r1<=r2<=...<=rl, the number of multi-chains \pi1<=\pi2<=...<=\pil in Armstrong's generalised non-crossing partitions poset, where the poset rank of \pii equals ri, and where the "block structure" of \pi1 is prescribed. We show that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result is even new 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 Dn generalised non-crossing partitions poset, which, in its turn, leads to a proof of Armstrong's F=M Conjecture in type Dn, thus completing a computational proof of the F=M Conjecture for all types (after the earlier case-free proof by Tzanaki ["Faces of generalized cluster complexes and noncrossing partitions"]).

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