Nearly Equal Distributions of the Rank and the Crank of Partitions

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 $m$. Bringmann and Mahlburg observed that $N(\leq m,n)\leq M(\leq m,n)\leq N(\leq m+1,n)$for$m<0$and$1\leq n\leq 100$. They also pointed out that these inequalities can be restated as the existence of a re-ordering ${\tau }_n$ on the set of partitions of $n$ such that $\mathrm{\mid }\text{crank}(\lambda )\mathrm{\mid }-\mathrm{\mid }\text{rank}({\tau }_n(\lambda ))\mathrm{\mid }=0$ or $1$ for all partitions $\lambda$ of $n$, that is, the rank and the crank are nearly equal distributions over partitions of $n$. In the study of the spt-function, Andrews, Dyson and Rhoades proposed a conjecture on the unimodality of the spt-crank, and they showed that this conjecture is equivalent to the inequality $N(\le m,n)\le M(\le m,n)$ for $m<0$ and $n\ge 1$. We proved this conjecture by combiantorial arguments. In this paper, we prove the inequality $N(\le m,n)\le M(\le m,n)$ for $m<0$ and $n\ge 1$. Furthermore, we define a re-ordering ${\tau }_n$ of the partitions $\lambda$ of $n$ and show that this re-ordering ${\tau }_n$ leads to the nearly equal distribution of the rank and the crank. Using the re-ordering ${\tau }_n$, we give a new combinatorial interpretation of the function ospt$(n)$ defined by Andrews, Chan and Kim, which immediately leads to an upper bound for $ospt(n)$ due to Chan and Mao.