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).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. E. Andrews: The Theory of Partitions. Addison-Wesley, Reading, 1976.
N. G. De Bruijn: Polya's theory of counting. Appl. Combin. Math. (E. F. Beckenbach, ed.). Wiley, New York, 1964, pp. 144–184.
P. Flajolet and M. Soria: Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theory 53 (1990), 165–182.
F. Harary and E. M. Palmer: Graphical Enumeration. Academic Press, New York-London, 1973.
D. E. Knuth: A note on solid partitions. Math. Comp. 24 (1970), 955–961.
G. Polya: Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und chemische Verbindungen. Acta Math. 68 (1937), 145–254.
G. Polya and R. C. Read: Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, New York, 1987.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wu, X., Chao, CY. An Application of Polya's Enumeration Theorem to Partitions of Subsets of Positive Integers. Czech Math J 55, 611–623 (2005). https://doi.org/10.1007/s10587-005-0049-2
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10587-005-0049-2