Partially Ordinal Sums and $P$-partitions

Abstract

We present a method of computing the generating function $f_P(\textbf{x})$ of $P$-partitions of a poset $P$. The idea is to introduce two kinds of transformations on posets and compute $f_P(\textbf{x})$ by recursively applying these transformations. As an application, we consider the partially ordinal sum $P_n$ of $n$ copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the sequence $\{f_{P_n}(\textbf{x})\}_{n\ge 1}$ 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.