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On the polynomial representation for the number of partitions with fixed length


Authors: So Ryoung Park, Jinsoo Bae, Hyun Gu Kang and Iickho Song
Journal: Math. Comp. 77 (2008), 1135-1151
MSC (2000): Primary 05A17; Secondary 11P81, 11P82
DOI: https://doi.org/10.1090/S0025-5718-07-02082-0
Published electronically: December 10, 2007
MathSciNet review: 2373195
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Abstract: In this paper, it is shown that the number $ M(n,k)$ of partitions of a nonnegative integer $ n$ with $ k$ parts can be described by a set of $ \widetilde{k}$ polynomials of degree $ k-1$ in $ Q_{\widetilde{k}}$, where $ \widetilde{k}$ denotes the least common multiple of the $ k$ integers $ 1, 2, \cdots, k$ and $ Q_{\widetilde{k}}$ denotes the quotient of $ n$ when divided by $ \widetilde{k}$. In addition, the sets of the $ \widetilde{k}$ polynomials are obtained and shown explicitly for $ k=3, 4, 5,$ and $ 6$.


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Additional Information

So Ryoung Park
Affiliation: School of Information, Communications, and Electronics Engineering, The Catholic University of Korea, Bucheon 420-743 Korea
Email: srpark@catholic.ac.kr

Jinsoo Bae
Affiliation: Department of Information and Communication Engineering, Sejong University, Seoul 143-747 Korea
Email: baej@sejong.ac.kr

Hyun Gu Kang
Affiliation: Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701 Korea
Email: khg@Sejong.kaist.ac.kr

Iickho Song
Affiliation: Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701 Korea
Email: i.song@ieee.org

DOI: https://doi.org/10.1090/S0025-5718-07-02082-0
Keywords: Partition, polynomial representation, nonrecursive formula
Received by editor(s): March 9, 2007
Published electronically: December 10, 2007
Additional Notes: This study was supported by the National Research Laboratory (NRL) Program of Korea Science and Engineering Foundation (KOSEF), Ministry of Science and Technology (MOST), under Grant R0A-2005-000-10005-0, for which the authors would like to express their thanks. The authors also wish to express their appreciation of the constructive suggestions and helpful comments from the anonymous reviewers.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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