Uniform asymptotic formulas of restricted bipartite partitions

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.