Partitions with k sizes from a set

A. David Christopher
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 1, Pages 100–108
DOI: 10.7546/nntdm.2022.28.1.100-108
Full paper (PDF, 202 Kb)

Details

Authors and affiliations

A. David Christopher 
Department of Mathematics, The American College
Tamil Nadu, India

Abstract

Let and be two positive integers and let be a set of positive integers. We define to be the number of partitions of with exactly sizes and parts in . As an implication of a variant of Newton’s product-sum identities we present a generating function for . Subsequently, we obtain a recurrence relation for and a divisor-sum expression for . Also, we present a bijective proof for the latter expression.

Keywords

• Newton’s product-sum identities
• Size of a partition
• Recurrence relation

2020 Mathematics Subject Classification

• Primary 05A17
• Secondary 11P99

References

1. Andrews, G. E. (1999). Stacked lattice boxes. Annals of Combinatorics, 3(2), 115–130.
2. Benyahia Tani, N., & Bouroubi, S. (2011). Enumeration of the partitions of an
   integer into parts of a specified number of different sizes and especially two sizes.
   Journal of Integer Sequences, 14, Article 11.3.6.
3. David Christopher, A. (2015). Partitions with fixed number of sizes. Journal of Integer Sequences, 18, Article 15.11.5.
4. Keith, W. J. (2017). Partitions into a small number of part sizes. International Journal of Number Theory, 13(1), 229–241.
5. Riordan, J. (1958). Introduction to Combinatorial Analysis, John Wiley & Sons, Inc., New York (1958).
6. Zeilberger, D. (1984). A combinatorial proof of Newton’s identities. Discrete Mathematics, 49(3), 319

Manuscript history

• Received: 19 October 2020
• Revised: 1 October 2021
• Accepted: 18 February 2022
• Online First: 19 February 2022