DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS AND GENERALISED NON-CROSSING PARTITIONS

Given a finite irreducible Coxeter group W, a positive integer d, and types T-1, T-2, ... , T-d (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c = sigma(1)sigma(2) ... 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, ... , 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, ... , 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 [Seminaire Lotharingien Combin. 54 (2006), Article B541]. 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 Bona, 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) <= r(2) <= ... <= r(l), the number of multi-chains pi(1) <= pi(2) <= ... <= 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.