##### This material has been published in
Séminaire
Lotharingien Combin. **54** (2006), Article B54l, 34 pp.

## Christian Krattenthaler

# The *M*-triangle of generalised non-crossing partitions
for the types *E*_{7} and *E*_{8}

### (34 pages)

**Abstract.**
The *M*-triangle of a ranked locally finite
poset *P* is the generating function *\sum
_{u,w \in P} \mu*(*u*,*w*)
*x*^{rk u} *y*^{rk w},
where *\mu*(.,.) 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 *E*_{7} and
*E*_{8}. For the other types except
*D*_{n} this had been
accomplished in the earlier paper
``The *F*-triangle of the
generalised cluster complex."
Altogether, this almost settles Armstrong's
*F*=*M* Conjecture predicting a surprising relation between
the *M*-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 *D*_{n}.
Moreover, we prove
a reciprocity result for this *M*-triangle,
again with the possible exception of type *D*_{n}.
Our results are based on the calculation of
certain decomposition numbers for the
reflexion groups of types *E*_{7} and *E*_{8}, which carry in fact finer
information than does the *M*-triangle. The decomposition numbers for the
other exceptional reflexion groups had been computed in the earlier
paper.
We present a conjectured formula for the type *A*_{n} decomposition numbers.

Here are the *Mathematica*
inputs for the decomposition number computations reported
in the paper.

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