A combinatorial proof of a plethystic Murnaghan--Nakayama rule

TL;DR

This paper provides a combinatorial proof for a generalized Murnaghan--Nakayama rule involving plethysm, expressing certain products of Schur functions as linear combinations, using bead move involutions on James' abacus.

Contribution

It introduces a new combinatorial proof for a plethystic generalization of the Murnaghan--Nakayama rule, expanding understanding of Schur function identities.

Findings

Expresses product of Schur function with plethysm as a linear combination of Schur functions

Uses sign-reversing involution on bead move sequences on James' abacus

Provides a combinatorial proof inspired by previous abacus arguments

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.