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. 



complete estimates of the error for a well-known asymptotic expression of partition p(n,m)
I'm not entirely sure of what you are asking, but note that Erdos and Lehner proved [here](https://projecteuclid.org/journals/duke-mathematical-journal/volume-8/issue-2/The-distribution-of-the-number-of-summands-in-the-partitions/10.1215/S0012-7094-41-00826-8.short) that
$$p(n,m)\sim \frac{n^{m-1}}{m!(m-1)!}$$ holds for $m=o(n^{1/3})$. In generality for any finite set $A$, with $|A|=m$ and $p(n,A)$ denoting the number of partitions of $n$ with parts from $A$, one has
$$p(n,A)=\frac{1}{\prod\_{a\in A}a}\frac{n^{m-1}}{(m-1)!}+O(n^{m-2}).$$


Such estimations can be deduced from the generating function of $p$ by using methods that are described in many books, for example "Analytic Combinatorics" by Flajolet and Sedgewick.


