The $(P, ω)$-partition generating function of a labeled poset $(P, ω)$ 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 $\{ψ_α\}$. 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 $ψ_α$-expansion of the $(P, ω)$-partition generating function akin to the Murnaghan-Nakayama rule.