P-partitions and quasi-symmetric functions.
Peter McNamara (LACIM, UQAM)
Given a partially ordered set (poset) P and a labelling
of its vertices,
we will give a definition of a P-partition, as introduced by Richard
Stanley in his Ph.D. thesis. In this thesis, Stanley made a conjecture
concerning a certain quasi-symmetric generating function for the set of
P-partitions of a labelled poset. This conjecture, which remains open,
says that the generating function is a symmetric function if and only if
our labelled poset is a "Schur labelled skew shape poset." In 1995,
Claudia Malvenuto reformulated the conjecture so that the symmetry of the
generating function needs to be related only to the local structure of the
labelled poset, rather than its global structure. We will discuss a
generalization of the idea of a P-partition, an appropriate extension of
Stanley's conjecture, and an extension of Malvenuto's reformulation.
We
will also explain how Stanley's conjecture is almost always true and
discuss several open problems concerning these quasi-symmetric generating
functions.
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