Proof of the plethystic Murnaghan-Nakayama rule using Loehr's labelled abacus

Abstract

The plethystic Murnaghan-Nakayama rule describes how to decompose the product of a Schur function and a plethysm of the form $p_r\circ h_m$ as a sum of Schur functions. We provide a short, entirely combinatorial proof of this rule using the labelled abaci introduced in Nicholas A. Loehr. Abacus proofs of Schur function identities. In: SIAM J. Discrete Math. 24.4 (2010), pp. 1356-1370.

12 pages, 4 figures. v2: Minor referee corrections. v3: Minor typos corrected