A note on Andrews-MacMahon theorem

TL;DR

This paper discusses a generalization of Andrews' theorem relating partitions with odd multiplicities to those with odd parts in specific residue classes, aiming to provide a bijective proof in the style of previous work.

Contribution

It introduces a new bijective mapping for the extended version of Andrews' theorem, aligning with the style of earlier bijections by Andrews, Ericksson, Petrov, and Romik.

Findings

Established a generalized bijection for the extended theorem

Connected partition sets with different combinatorial constraints

Enhanced understanding of partition identities and bijections

Abstract

For a positive integer $r$, George Andrews proved that the set of partitions of $n$ in which odd multiplicities are at least $2r + 1$ is equinumerous with the set of partitions of $n$ in which odd parts are congruent to $2r + 1$ modulo $4r + 2$. This was given as an extension of MacMahon's theorem ($r = 1$). Andrews, Ericksson, Petrov and Romik gave a bijective proof of MacMahon's theorem. Despite several bijections being given, until recently, none of them was in the spirit of Andrews-Ericksson-Petrov-Romik bijection. Andrews' theorem has also been extended recently. Our goal is to give a generalized bijective mapping of this further extension in the spirit of Andrews-Ericksson-Petrov-Romik bijection.