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:
$q(n,M)\approx \frac{(n-1)!}{M!(M-1)!(n-N)!}\left( 1+O\left( \frac{M^{3}}{n}$
Isn't there no similar asymptotic expression for partition $p(n,m)$?