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A Unified Approach to Algorithms Generating Unrestricted and Restricted Integer Compositions and Integer Partitions

Journal of Mathematical Modelling and Algorithms

36 Pages Posted: 18 Aug 2008 Last revised: 31 Jan 2011

J.D. Opdyke

DataMineit, LLC; GE Capital

Date Written: August 16, 2008

Abstract

An original algorithm is presented that generates both restricted integer compositions and restricted integer partitions that can be constrained simultaneously by a) upper and lower bounds on the number of summands (“parts”) allowed, and b) upper and lower bounds on the values of those parts. The algorithm can implement each constraint individually, or no constraints to generate unrestricted sets of integer compositions or partitions. The algorithm is recursive, based directly on very fundamental mathematical constructs, and given its generality, reasonably fast with good time complexity. A general, closed form solution to the open problem of counting the number of integer compositions doubly restricted in this manner also is presented; its formulaic link to an analogous solution for counting doubly-restricted integer partitions is shown to mirror the algorithmic link between these two objects.

Keywords: Integer Compositions, Integer Partitions, Bounded Compositions, Bounded Partitions, Pascal's triangle, Fibonacci

JEL Classification: C02, C65, C88

Suggested Citation

Opdyke, J.D., A Unified Approach to Algorithms Generating Unrestricted and Restricted Integer Compositions and Integer Partitions (August 16, 2008). Journal of Mathematical Modelling and Algorithms. Available at SSRN: https://ssrn.com/abstract=1231502

J.D. Opdyke (Contact Author)

DataMineit, LLC; GE Capital ( email )

FL
United States
617-943-6463 (Phone)
781-639-6463 (Fax)

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