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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 $1leq nleq 100$. 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 $ngeq 1$. 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 $ngeq 1$. 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.
A generalized crank ($k$-crank) for $k$-colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined $k$-crank. Namely, we first obtain some inequalities between the $k$-crank counts
The problem of finding completely positive matrices with equal cp-rank and rank is considered. We give some easy-to-check sufficient conditions on the entries of a doubly nonnegative matrix for it to be completely positive with equal cp-rank and rank
We consider $m$-divisible non-crossing partitions of ${1,2,ldots,mn}$ with the property that for some $tleq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing par
If L is a partition of n, the rank of L is the size of the largest part minus the number of parts. Under the uniform distribution on partitions, Bringmann, Mahlburg, and Rhoades showed that the rank statistic has a limiting distribution. We identify
We derive the asymptotic formula for $p_n(N,M)$, the number of partitions of integer $n$ with part size at most $N$ and length at most $M$. We consider both $N$ and $M$ are comparable to $sqrt{n}$. This is an extension of the classical Hardy-Ramanuja