On the distribution of rank and crank statistics for integer partitions

TL;DR

This paper derives asymptotic formulas for Garvan's k-rank distribution in integer partitions, providing insights into Dyson's crank conjecture and analyzing finite differences for large m.

Contribution

It offers new asymptotic results for Garvan's k-rank distribution and advances understanding of Dyson's crank conjecture for large parameters.

Findings

Asymptotic formulas for N_k(m,n) when |m| ≥ √n as n→∞

Refined understanding of Dyson's crank distribution conjecture

Asymptotics for finite differences of N_k(m,n) with large m

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 $m\gg \sqrt{n}\log n$.