Two-color partitions and overpartitions: a combinatorial proof

TL;DR

This paper provides a combinatorial proof connecting two-color partitions and overpartitions, addressing conjectures made by Andrews and El Bachraoui through modular diagram techniques.

Contribution

It introduces a novel combinatorial proof using two-modular diagrams for identities originally proven analytically by Andrews and El Bachraoui.

Findings

Established a combinatorial proof for the main identity

Confirmed the conjectured connection between two-color partitions and overpartitions

Enhanced understanding of partition identities through modular diagrams

Abstract

George Andrews and Mohamed El Bachraoui recently explored identities for two-color partitions. In particular, they studied the connection between two-colored partitions and overpartitions. Their proofs were analytical, but they conjectured combinatorial proofs of their results. In this paper we use two-modular diagrams to give a combinatorial proof of their main result.