# Asymptotic of the number of partitions of $n$ into numbers from $\{1, 2, \dots, k\}$

How to find the asymptotic behavior ($n \to +\infty$) of the number $q(n, k)$ of partitions of $n$ into addends from $\{1, 2, \dots, k\}$?

I proved that $q(n, k)$ satisfies the recurrent relation $q(n, k) = q(n - k, k) + q(n, k - 1)$ for $n \geq k$. How can I find asymptotic then? Any help would be appreciated.

• Thanks, I read this article. There is written that If $A$ is a set of natural numbers which possesses positive natural density $\alpha$ and $p_A(n)$ denote the number of partitions of $n$ into elements of $A$, then $p_A(n) \sim C\sqrt{\alpha n}$ for $C = \pi\sqrt{\dfrac{2}{3}}$. However, in this case the natural density of $\{1, 2, \dots, k\}$ is 0, so it doesn't work, does it? – Victor Mar 7 '16 at 13:20