Set partitions with isolated successions

Toufik Mansour and Augustine O. Munagi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 1, Pages 81—91
DOI: 10.7546/nntdm.2022.28.1.81-91
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Authors and affiliations

Toufik Mansour
Department of Mathematics, University of Haifa
3498838 Haifa, Israel

Augustine O. Munagi
School of Mathematics, University of the Witwatersrand
Johannesburg 2050, South Africa


We enumerate partitions of the set \{1,\dots,n\} according to occurrences of isolated successions, that is, integer strings a,a+1,\dots,b in a block when neither a-1 nor b+1 lies in the same block. Our results include explicit formulas and generating functions for the number of partitions containing isolated successions of a given length.
We also consider a corresponding analog of the associated Stirling numbers of the second kind.


  • Partition
  • Isolated succession
  • Recurrence
  • Generating function

2020 Mathematics Subject Classification

  • 05A18
  • 05A15
  • 05A19


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Manuscript history

  • Received: 24 March 2021
  • Revised: 14 February 2021
  • Accepted: 15 February 2022
  • Online First: 15 February 2022

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Cite this paper

Mansour, T., & Munagi, A. O. (2022). Set partitions with isolated successions. Notes on Number Theory and Discrete Mathematics, 28(1), 81-91, DOI: 10.7546/nntdm.2022.28.1.81-91.

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