P-Partitions and Quasisymmetric Power Sums

Abstract

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.