Let $p(n,m)$ be the number of partitions of an integer $n$ into integers $\le m$, we have a well-known asymptotic expression:

For a fixed $m$ and $n\to\infty$, $$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+O(1/n)) $$

My question is: why the error $O(1/n)$ is independent of $m$? Or how can it be extended for $m$ growing slowly with $n$? Please help me to find the answer or the references. Thanks.

Theory of Partitions. At the very least you might check there. (This is a comment instead of an answer because I don't actually have the book in front of me to check.) – Michael Lugo Sep 1 '10 at 15:14