Uniform Asymptotics for Andrews-Beck Partition Statistics

Let $\text{NT}(m,n)$ be the total number of parts in the partitions of $n$ with rank $m$, and let $\text{MW}(m,n)$ 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 $\text{MW}(m,n)$. Put ${\beta }_n=\pi \mathrm{/}\sqrt{6n}$ and $L_n=\log (1\mathrm{/}{\beta }_n)$. For every fixed $A>0$, uniformly for $\mathrm{\mid }m\mathrm{\mid }\le A\sqrt{n}$, we prove \begin{align*} \NT(m,n)&=\frac{p(n)L_n}{4}\sech^2!\left(\frac{β_nm}{2}\right) +O_A!\bigl(p(n)\log L_n\bigr) \intertext{ and} \MW(m,n)&=\frac{p(n)}4\log(1+e^{-β_nm}) \sech^2!\left(\frac{β_nm}{2}\right) +O_A!\bigl(p(n)\sqrt{β_nL_n}\bigr). \end{align*}