  Let $N(\leq m,n)$ denote the number of partitions of $n$ with rank not
greater than $m$, and let $M(\leq m,n)$ 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
$|\text{crank}(\lambda)|-|\text{rank}(\tau_n(\lambda))|=0$ 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
$N(\leq m,n)\leq M(\leq m,n)$ for $m<0$ and $n\geq 1$. We proved this
conjecture by combiantorial arguments. In this paper, we prove the inequality
$N(\leq m,n)\leq M(\leq m,n)$ for $m<0$ and $n\geq 1$. 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.
