Combinatorial proof of identities involving partitions with distinct even parts and 4-regular partitions

TL;DR

This paper provides combinatorial proofs for identities connecting partitions with distinct even parts to 4-regular partitions, complementing previous hypergeometric series-based proofs.

Contribution

It introduces bijections that offer the first combinatorial proofs for these identities, advancing understanding of partition theory.

Findings

Established bijections for the identities

Provided combinatorial proofs complementing hypergeometric series methods

Enhanced combinatorial understanding of partition identities

Abstract

Recently Andrews and Bachraoui proved identities relating certain restricted partitions into distinct even parts with restricted 4-regular partitions by the theory of basic hypergeometric series. They also posed a question regarding combinatorial proofs for these results. In this paper, we establish bijections to provide combinatorial proofs for these results.