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

Abstract

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.

Keywords

  • Partition
  • Isolated succession
  • Recurrence
  • Generating function

2020 Mathematics Subject Classification

  • 05A18
  • 05A15
  • 05A19

References

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  3. Heubach, S., & Mansour, T. (2010). Combinatorics of Compositions and Words. CRC, Taylor & Francis Group, Boca Raton.
  4. Mansour, T. (2013). Combinatorics of Set Partitions. CRC, Taylor & Francis Group, Boca Raton.
  5. Mansour, T., & Shattuck, M. (2013). Parity successions in set partitions. Linear Algebra and Its Applications, 439(9), 2642–2650.
  6. Munagi, A. O. (2008). Extended set partitions with successions. European Journal of Combinatorics, 29 (5), 1298–1308.
  7. Munagi, A. O. (2018). Set partitions with isolated singletons. The American Mathematical Monthly, 125(5), 447–452.
  8. Riordan, J. (1958). An Introduction to Combinatorial Analysis, John Wiley, New York.
  9. Shattuck, M. (2015). Combinatorial proofs of some Stirling number formulas. Pure Mathematics and Applications, 25(1), 107–113.

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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