Geodesic
An application of Pólya’s enumeration theorem to partitions of subsets of positive integers
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 611-623

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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 $a_1, a_2, \dots , a_r$ in $S$ with repetitions allowed such that $\sum ^r_{i=1} a_i = n$. Here we apply Pólya’s enumeration theorem to find the number $¶(n;S)$ of partitions of $n$ into $S$, and the number ${\mathrm DP}(n;S)$ of distinct partitions of $n$ into $S$. We also present recursive formulas for computing $¶(n;S)$ and ${\mathrm DP}(n;S)$.
Classification : 05A15, 05A17, 11P81
Keywords: Pólya’s enumeration theorem; partitions of a positive integer into a non-empty subset of positive integers; distinct partitions of a positive integer into a non-empty subset of positive integers; recursive formulas and algorithms 
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     author = {Wu, Xiaojun and Chao, Chong-Yun},
     title = {An application of {P\'olya{\textquoteright}s} enumeration theorem to partitions of subsets of positive integers},
     journal = {Czechoslovak Mathematical Journal},
     pages = {611--623},
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     volume = {55},
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     mrnumber = {2153086},
     zbl = {1081.05010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005__55_3_a3/}
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Wu, Xiaojun; Chao, Chong-Yun. An application of Pólya’s enumeration theorem to partitions of subsets of positive integers. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 611-623. http://geodesic.mathdoc.fr/item/CMJ_2005__55_3_a3/