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

TL;DR

This paper extends the Murnaghan-Nakayama rule to weighted $(P,w)$-partitions, providing a combinatorial proof and generalizing recent formulas for quasisymmetric functions from labeled posets.

Contribution

It introduces a weighted version of the Murnaghan-Nakayama rule for $(P,w)$-partition generating functions with a combinatorial proof, broadening the scope of previous results.

Findings

Extended the Murnaghan-Nakayama rule to weighted $(P,w)$-partitions

Provided a combinatorial proof avoiding Hopf algebra machinery

Generalized formulas for coefficients in quasisymmetric power sum basis

Abstract

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.