An Application of Polya's Enumeration Theorem to Partitions of Subsets of Positive Integers

Abstract

Let S be a non-empty subset of positive integers. A partition of a positive integer n into S is a finite nondecreasing sequence of positive integers a 1, a 2,...,a r in S with repetitions allowed such that $$\sum\limits_{i = 1}^r {a_i = n}$$ . Here we apply Polya's enumeration theorem to find the number P(n; S) of partitions of n into S, and the number DP(n; S) of distinct partitions of n into S. We also present recursive formulas for computing P(n; S) and DP(n; S).