Uniform Asymptotics for Andrews-Beck Partition Statistics

Abstract

Let $\NT(m,n)$ be the total number of parts in the partitions of $n$ with rank $m$, and let $\MW(m,n)$ be the total number of ones in the partitions of $n$ with crank $m$. Using the uniform generalized-rank estimate of Liu and Zhou, we establish asymptotic formulas for $NT(m,n)$ and $\MW(m,n)$. Put $\beta_n=\pi/\sqrt{6n}$ and $L_n=\log(1/\beta_n)$. For every fixed $A>0$, uniformly for $|m|\le A\sqrt n$, we prove \begin{align*} \NT(m,n)&=\frac{p(n)L_n}{4}\sech^2\!\left(\frac{\beta_nm}{2}\right) +O_A\!\bigl(p(n)\log L_n\bigr) \intertext{ and} \MW(m,n)&=\frac{p(n)}4\log(1+e^{-\beta_nm}) \sech^2\!\left(\frac{\beta_nm}{2}\right) +O_A\!\bigl(p(n)\sqrt{\beta_nL_n}\bigr). \end{align*}