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arXiv · 1803.11081

On the distribution of rank and crank statistics for integer partitions

Nian Hong Zhou
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Abstract

Let $k$ be a positive integer and $m$ be an integer. Garvan's $k$-rank $N_k(m,n)$ is the number of partitions of $n$ into at least $(k-1)$ successive Durfee squares with $k$-rank equal to $m$. In this paper give some asymptotics for $N_k(m,n)$ with $|m|\ge \sqrt{n}$ as $n\rightarrow \infty$. As a corollary, we give a more complete answer for the Dyson's crank distribution conjecture. We also establish some asymptotic formulas for finite differences of $N_k(m,n)$ with respect to $m$ with $m\gg \sqrt{n}\log n$.

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Nian Hong Zhou. 2018-03-29. On the distribution of rank and crank statistics for integer partitions. https://doi.org/10.1007/s40993-019-0156-z

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