  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.
