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 E7 and E8
(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)
xrk u yrk 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 E7 and
E8. For the other types except
Dn 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 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
reflexion groups of types E7 and E8, 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 An decomposition numbers.
Here are the Mathematica
inputs for the decomposition number computations reported
in the paper.
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