A bijective proof of a generalization of the non-negative crank--odd mex identity

TL;DR

This paper provides a bijective proof of a generalized identity linking partition statistics, the crank and the mex, using an alternative Durfee decomposition approach to establish the combinatorial connection.

Contribution

It introduces a new bijective proof of a generalized crank-odd mex identity, expanding understanding of partition statistics through a novel combinatorial method.

Findings

Established a bijective proof of the generalized identity.

Linked Durfee decomposition to the crank statistic.

Extended previous identities to a broader class of partitions.

Abstract

Recent works of Andrews--Newman and Hopkins--Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer $n$, there are as many partitions of $n$ with non-negative crank as partitions of $n$ with odd mex. In this paper, we give a bijective proof of a generalization of this identity provided by Hopkins, Sellers, and Stanton. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee.