A refinement of and a companion to MacMahon's partition identity

TL;DR

This paper refines MacMahon's partition identity, introduces a new mod 6 partition identity, and extends the technique to cover Andrews's generalization, using bijective proofs based on a theorem by Xiong and Keith.

Contribution

It provides a refined version of MacMahon's identity and extends the approach to a broader class of partition identities, including Andrews's generalization.

Findings

Refined MacMahon's partition identity.

Derived a new mod 6 partition identity.

Extended the technique to Andrews's generalization.

Abstract

We provide a refinement of MacMahon's partition identity on sequence-avoiding partitions, and use it to produce another mod 6 partition identity. In addition, we show that our technique also extends to cover Andrews's generalization of MacMahon's identity. Our proofs are bijective in nature, exploiting a theorem of Xiong and Keith.