Combinatorial Extensions on Andrews and El Bachraoui's Almost-Distinct Partitions

TL;DR

This paper provides combinatorial proofs for a family of identities related to almost-distinct partitions, extending previous work and exploring inequalities for partitions with specific repetition constraints.

Contribution

It introduces combinatorial proofs for an infinite family of identities and examines partitions where only the largest part is repeated, expanding understanding of almost-distinct partitions.

Findings

Proved infinite families of identities for almost-distinct partitions.

Derived inequalities for the count of partitions with specific repetition rules.

Extended previous results to new classes of partitions.

Abstract

Andrews and El Bachraoui recently studied integer partitions where the smallest part is repeated a specified number of times and any other parts are distinct. Their results included two ``surprising identities'' for which they requested combinatorial proofs. We provide combinatorial proofs for an infinite family of related identities and also consider the analogous partitions where only the largest part can be repeated. The results give rise to infinite families of inequalities for the number of partitions with distinct parts.