Proof of the Plethystic Murnaghan–Nakayama Rule Using Loehr’s Labelled Abacus

The plethystic Murnaghan–Nakayama rule describes how to decompose the product of a Schur function and a plethysm of the form \(p_r\circ h_m\) $p_r\circ h_m$ as a sum of Schur functions. We provide a short, entirely combinatorial proof of this rule using the labelled abaci introduced in Loehr (SIAM J Discrete Math 24(4):1356–1370, 2010).