Combinatorial Proofs of Some Results of Andrews and El Bachraoui

TL;DR

This paper provides combinatorial proofs for recent $q$-series identities related to restricted partitions, extending the results to partitions into distinct odd parts, thereby offering new combinatorial insights.

Contribution

It introduces combinatorial proofs for identities originally proved analytically, and extends these identities to partitions into distinct odd parts.

Findings

Validated identities for restricted partitions into even parts

Extended identities to partitions into odd parts

Enhanced understanding of $q$-series and partition combinatorics

Abstract

Recently, Andrews and El Bachraoui (2024) proved three very interesting $q$-series identities, from which three simple looking identities involving certain restricted partitions into distinct even parts and $4$-regular partitions follow. In this short note, we give combinatorial proofs of these identities. We also prove the counterpart identities for the restricted partitions into distinct odd parts.