Nearly Equal Distributions of the Rank and the Crank of Partitions

William Y. C. Chen, Kathy Q. Ji, Wenston J. T. Zang

Let $N(\le m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(\le m,n)$ denote the number of partitions of $n$ with crank not greater than $m…