On asymptotic formula of the partition function $p_A(n)$

Abstract

The partition function, $p_A(n)$, is defined to be the number of partitions of $n$ with parts in the set A, where $n$ is a positive integer and $A$ is a set of positive integers. It is well documented that: if A is a finite set with $\gcd(A)=1$ and $|A|=k$, then \[p_A(n)\sim \frac{n^{k-1}}{(\prod_{a\in A}a)(k-1)!}. \] Number of proofs have been obtained for this estimate. In this article, we give a new proof for the above estimate by making use of the fact that: $p_A(n)$ is a $quasi\ polynomial$ when A is a finite set. Present method of proof is purely combinatorial.