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
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Authors and affiliations

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


Let n and k be two positive integers and let A be a set of positive integers. We define t_A(n,k) to be the number of partitions of n with exactly k sizes and parts in A. As an implication of a variant of Newton’s product-sum identities we present a generating function for t_A(n,k). Subsequently, we obtain a recurrence relation for t_A(n,k) and a divisor-sum expression for t_A(n,2). Also, we present a bijective proof for the latter expression.


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

2020 Mathematics Subject Classification

  • Primary 05A17
  • Secondary 11P99


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Manuscript history

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

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Cite this paper

David Christopher, A. (2022). Partitions with k sizes from a set. Notes on Number Theory and Discrete Mathematics, 28(1), 100-108, DOI: 10.7546/nntdm.2022.28.1.100-108.

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