P-partitions revisited

P-partitions for a partially ordered set P were introduced by Stanley around 1970, as a unification of several kinds of number partitions: weak partitions, strict partitions, plane partitions, column-strict tableaux, etc. One can view the main lemma on P-partitions as a triangulation of a certain polyhedral cone associated with P into unimodular subcones, indexed by the linear extensions of P.

By analyzing the structure of these cones, or rather the semigroup ring associated with their lattice points, we deduce some new results on P-partitions. This involves a different but equally natural triangulation of this cone, and in some cases, leads to new formulas enumerating linear extensions.