Partially Ordinal Sums and $P$-partitions

TL;DR

This paper introduces a recursive method for computing generating functions of P-partitions in posets, especially focusing on partially ordinal sums, and demonstrates its effectiveness through various examples.

Contribution

It develops a new recursive approach using transformations on posets to compute generating functions for P-partitions, including for complex constructions like partially ordinal sums.

Findings

The sequence of generating functions for partially ordinal sums satisfies finite recurrence relations.

The method is illustrated with examples including 3-rowed posets and multi-cube posets.

The approach generalizes previous methods for specific poset sums.

Abstract

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