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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 $a_1,a_2,\cdots ,a_r$ in $S$ with repetitions allowed such that ${\sum }_{i=1}^r{a}_i=n$. Here we apply Pólya’s enumeration theorem to find the number $\P (n;S)$ of partitions of $n$ into $S$, and the number $\mathrm{D}P(n;S)$ of distinct partitions of $n$ into $S$. We also present recursive formulas for computing $\P (n;S)$ and $\mathrm{D}P(n;S)$.