$P$-Partitions and Quasisymmetric Power Sums

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.