Partitions with parts in a finite set

Abstract:

Let $A$ be a nonempty finite set of relatively prime positive integers, and let $p_A(n)$ denote the number of partitions of $n$ with parts in $A$. An elementary arithmetic argument is used to prove the asymptotic formula \[ p_A(n) = \left (\frac {1}{\prod _{a\in A}a}\right ) \frac {n^{k-1}}{(k-1)!} + O\left ( n^{k-2}\right ). \]