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

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