Partially Ordinal Sums and P-partitions

We present a method of computing the generating function fP (x) of P -partitions of a poset P . The idea is to introduce two kinds of transformations on posets and compute fP (x) by recursively applying these transformations. As an application, we consider the partially ordinal sum Pn of n copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the sequence {fPn(x)}n>1 satisfies a finite system of recurrence relations with respect to n. We illustrate the method by several examples, including a kind of 3-rowed posets and the multi-cube posets.