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 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.