Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions
Transactions of the American Mathematical Society
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 = σ 1 σ 2 · · · σ d of a Coxeter element c of W , such that σ 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
... 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 ≤ r 2 ≤ · · · ≤ r l , the number of multi-chains π 1 ≤ π 2 ≤ · · · ≤ π l in Armstrong's generalised non-crossing partitions poset, where the poset rank of π i equals r i and where the "block structure" of π 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.