Séminaire Lotharingien de Combinatoire, B54l (2006), 34 pp.
The M-Triangle of
Generalised Non-Crossing Partitions
for the Types E7 and E8
The M-triangle of a ranked locally finite
poset P is the generating function \sum
u,w \in P \mu(u,w)
xrk uyrk 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"
[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 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.
Received: July 27, 2006.
Revised: September 19, 2006.
Accepted: October 6, 2006.
The following versions are available: