Partitions with parts in a finite set

Abstract

LetAAbe a nonempty finite set of relatively prime positive integers, and letpA(n)p_A(n)denote the number of partitions ofnnwith parts inAA. An elementary arithmetic argument is used to prove the asymptotic formula\[pA(n)=(1∏a∈Aa)nk−1(k−1)!+O(nk−2).p_A(n) = \left (\frac {1}{\prod _{a\in A}a}\right ) \frac {n^{k-1}}{(k-1)!} + O\left ( n^{k-2}\right ).\]