Combinatorial proofs for partitions with repeated smallest part

TL;DR

This paper provides combinatorial bijections to prove identities related to integer partitions where the smallest part is repeated exactly k times, addressing questions posed by Andrews and El Bachraoui.

Contribution

It introduces new combinatorial bijections that serve as proofs for identities involving partitions with a repeated smallest part, filling a gap in existing combinatorial proofs.

Findings

Established bijections for partitions with repeated smallest parts

Provided combinatorial proofs for identities posed by Andrews and El Bachraoui

Enhanced understanding of partition structures with specific repetition constraints

Abstract

Recently, Andrews and El Bachraoui considered the number of integer partitions whose smallest part is repeated exactly $k$ times and the remaining parts are not repeated. They presented several interesting results and posed questions regarding combinatorial proofs for these identities. In this paper, we establish bijections to provide combinatorial proofs for these results.