Decomposition Numbers for Finite Coxeter Groups and Generalised Non-crossing Partitions

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 = σ1σ2 · · ·σd of a Coxeter element c of W , such that σi 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 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 An 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 Bn decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. 24 (2000), 22–56] on map enumeration. Our formula for the typeDn 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 π1 ≤ π2 ≤ · · · ≤ πl in Armstrong’s generalised non-crossing partitions poset, where the poset rank of πi equals ri and where the “block structure” of π1 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 Dn generalised non-crossing partitions poset, which, in 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. 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.