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Séminaire Lotharingien de Combinatoire, B54l (2006), 34 pp.

# Christian Krattenthaler

# The *M*-Triangle of
Generalised Non-Crossing Partitions
for the Types *E*_{7} and *E*_{8}

**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"
[in: "Topics in Discrete Mathematics," M. Klazar, J. Kratochvil, M. Loebl,
J. Matousek, 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
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
reflection groups of types *E*_{7} and *E*_{8},
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 *A*_{n}
decomposition numbers, leaving the problem of computing the type *B*_{n}
and type *D*_{n} decomposition numbers open.

Received: July 27, 2006.
Revised: September 19, 2006.
Accepted: October 6, 2006.

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