The M - triangle of generalised non - crossing partitions for the types

The M -triangle of a ranked locally finite poset P is the generating function P u,w∈P μ(u,w)x rk uyrk w, where μ(., .) is the Möbius function of P . We compute the M triangle of Armstrong’s poset of m-divisible non-crossing partitions for the root systems of type E7 and E8. For the other types exceptDn this had been accomplished in the earlier paper “The F -triangle of the generalised cluster complex” [in: “Topics in Discrete Mathematics,” M. Klazar, J. Kratochvil, M. Loebl, J. Matoušek, R. Thomas and P. Valtr, eds., Springer– Verlag, Berlin, New York, 2006, pp. 93–126]. Altogether, this almost settles Armstrong’s F =M Conjecture, predicting a surprising relation between theM -triangle of the m-divisible partitions poset and the F -triangle (a certain refined face count) of the generalised cluster complex of Fomin and Reading, the only gap remaining in type Dn. Moreover, we prove a reciprocity result for this M -triangle, again with the possible exception of type Dn. Our results are based on the calculation of certain decomposition numbers for the reflection groups of types E7 and E8, which carry in fact finer information than the M -triangle does. The decomposition numbers for the other exceptional reflection groups had been computed in the earlier paper. As an aside, we show that there is a closed form product formula for the type An decomposition numbers, leaving the problem of computing the type Bn and type Dn decomposition numbers open.