##### This material has been published in
Trans. Amer. Math. Soc. **362** (2010), 2723-2787,
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# 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
*T*_{1},*T*_{2},...,*T*_{d} (in the sense of the classification of finite
Coxeter groups), we compute the number of
decompositions *c*=*\si*_{1}*\si*_{2}...*\si*_{d}
of a Coxeter element *c* of
*W*, such that *\si*_{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 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 *A*_{n} decomposition
numbers had been computed by Goulden and Jackson in [*Europ. J.
Combin.* **13** (1992), 357-365], and that the type *B*_{n}
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 *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 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 *D*_{n} generalised non-crossing partitions poset, which, in
its 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 (after the earlier case-free proof by Tzanaki
[*"Faces of generalized cluster
complexes and noncrossing partitions"*]).

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