Generalized (P,ω)-partitions and generating functions for trees

We introduce (P,R)-partitions as a generalization of the (P,ω)-partitions of Stanley. When P is a Gaussian poset the generating function for P-partitions with largest part at most n factors as ∏x∈P 1−qg(x)+n1−qg(x) for certain integers g(x). Although trees are not in general Gaussian posets, we show that if P is a tree then R can be chosen so that the generating function for (P,R)-partitions has a similar factorization.