# $P$-Partitions and Quasisymmetric Power Sums

## TL;DR

This paper explores the expansion of $P$-partition generating functions in the quasisymmetric power sum basis, revealing new structural properties and combinatorial interpretations related to labeled posets.

## Contribution

It introduces the expansion of $P$-partition functions in the quasisymmetric power sum basis and establishes their irreducibility and uniqueness properties for certain posets.

## Key findings

- Connected, naturally labeled posets have irreducible $P$-partition generating functions.
- Series-parallel posets are uniquely determined by their generating functions.
- Provides a combinatorial interpretation for the coefficients in the basis expansion.

## Abstract

The $(P, \omega)$-partition generating function of a labeled poset $(P, \omega)$ is a quasisymmetric function enumerating certain order-preserving maps from $P$ to $\mathbb{Z}^+$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $\{\psi_\alpha\}$. Using this expansion, we show that connected, naturally labeled posets have irreducible $P$-partition generating functions. We also show that series-parallel posets are uniquely determined by their partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients of the $\psi_\alpha$-expansion of the $(P, \omega)$-partition generating function akin to the Murnaghan-Nakayama rule.