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

Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,\dots,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\sigma_1\sigma_2\cdots\sigma_d$ 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 [``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 $r_1\le r_2\le \dots\le r_l$, the number of multi-chains $\pi_1\le \pi_2\le \dots\le \pi_l$ 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.