A combinatorial proof of a plethystic Murnaghan--Nakayama rule

Abstract

This article gives a 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. Loehr, Abacus proofs of Schur function identities, SIAM J. Discrete Math. 24 (2010), 1356-1370.