Uniform asymptotic formulas of restricted bipartite partitions

TL;DR

This paper derives uniform asymptotic formulas for the number of bipartite partitions into decreasing parts, relating it to the crank statistic, and provides new insights into their asymptotic behavior.

Contribution

It establishes uniform asymptotic formulas for bipartite partitions and links them to the crank statistic, advancing understanding of their distribution.

Findings

Derived uniform asymptotic formulas for $\pi(m,n)$

Established a relation between $\pi(m,n)$ and the crank statistic $M(m,n)$

Enhanced understanding of the asymptotic behavior of bipartite partitions

Abstract

In this paper, we investigate $\pi(m,n)$, the number of partitions of the \emph{bipartite number} $(m,n)$ into \emph{steadily decreasing} parts, introduced by L.Carlitz ['A problem in partitions', Duke Math Journal 30 (1963), 203--213]. We give a relation between $\pi(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, some uniform asymptotic formulas for $\pi(m,n)$ are established.