Thank Robin Chapman very much for editing.

There is a nice asymptotic expression for partition $q(n,M)$ that denotes the number of partitions of $n$  with $M$ parts all distinct: As $n\to\infty$,

$$ q(n,M)\approx \frac{(n-1)!}{M!(M-1)!(n-M)!}\left( 1+O\left( \frac{M^{3}}{n} \right) \right)$$


Isn't there no similar asymptotic expression for partition $p(n,m)$?