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In this paper, we investigate 
$\unicode[STIX]{x1D70B}(m,n)$, the number of partitions of the bipartite number
$(m,n)$ into steadily decreasing parts, introduced by Carlitz [‘A problem in partitions’, Duke Math. J.30 (1963), 203–213]. We give a relation between 
$\unicode[STIX]{x1D70B}(m,n)$ and the crank statistic 
$M(m,n)$ for integer partitions. Using this relation, we establish some uniform asymptotic formulas for 
$\unicode[STIX]{x1D70B}(m,n)$.
