Partially ordinal sums and \(P\)-partitions (Q1953336)

Summary: We present a method of computing the generating function \(f_P(\text{\textbf{x}})\) of \(P\)-partitions of a poset \(P\). The idea is to introduce two kinds of transformations on posets and compute \(f_P(\text{\textbf{x}})\) 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 \(\{f_{P_n}(\text{\textbf{x}})\}_{n\geq 1}\) 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.