A Combinatorial Proof of a Plethystic Murnaghan--Nakayama Rule

Abstract

This article gives a short combinatorial proof of a plethystic generalization of the Murnaghan--Nakayama rule. The main result expresses the product of a Schur function with the plethysm $p_r \circ h_n$ as an integral linear combination of Schur functions. The proof uses a sign-reversing involution on sequences of bead moves on James' abacus, inspired by the arguments in N. A. Loehr [SIAM J. Discrete Math., 24 (2010), pp. 1356--1370].