  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.
