A Bijective Proof of a Generalization of the Non-Negative Crank-Odd Mex Identity

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.