A weighted Murnaghan-Nakayama rule for $(P,w)$-partitions

The $(P,w)$-partition generating function $K_{(P,w)}(x)$ is a quasisymmetric function obtained from a labeled poset. Recently, Liu and Weselcouch gave a formula for the coefficients of $K_{(P,w)}(x)$ when expanded in the quasisymmetric power sum function basis. This formula generalizes the classical Murnaghan–Nakayama rule for Schur functions. We extend this result to weighted $(P,w)$-partitions and provide a short combinatorial proof, avoiding the Hopf algebra machinery used by Liu–Weselcouch.