A combinatorial proof of the Alladi-Gordon key identity for Schur's partition theorem

TL;DR

This paper provides a combinatorial proof of the Alladi-Gordon key identity related to Schur's partition theorem, using overpartition interpretation and an involution method.

Contribution

It introduces an overpartition interpretation and constructs an involution to give a new combinatorial proof of the identity.

Findings

Overpartition interpretation of the Alladi-Gordon identity

Construction of an involution on overpartitions

New combinatorial proof of the key identity

Abstract

The Alladi-Gordon identity plays an important role for the Alladi-Gordon generalization of Schur's partition theorem. By using Joichi-Stanton's insertion algorithm, we present an overpartition interpretation for the Alladi-Gordon key identity. Based on this interpretation, we further obtain a combinatorial proof of the Alladi-Gordon key identity by establishing an involution on the underlying set of overpartitions.