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Set partitions and partitions without singleton blocks of type B

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Abstract

In this paper we study the counting sequences which arise from the enumeration problems of set partitions of type B. These sequences count the total number of set partitions, the number of partitions with a given number of blocks. We thoroughly study the enumeration problem of partitions without singletons.

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References

  1. T. Amdeberhan, V. De Angelis, V. Moll, Complementary Bell numbers: Arithmetical properties and Wilf’s conjecture, in Advances in Combinatorics. In memory of Herbert S. Wilf. ed. by I.S. Kotsireas, E. Zima (Springer-Verlag, Berlin, 2013), pp. 23–56

  2. E. Bagno, R. Biagioli, and D. Garber. Some identities involving second kind Stirling numbers of types \(B\) and \(D\). Electron. J. Combin. 26(3) (2019) # P3.9, 1–20

  3. E. Bagno and D. Garber. Signed partitions - A balls into urns approach. (2019) arXiv:1903.02877

  4. P. Barry. On a family of generalized Pascal triangles defined by exponential Riordan arrays. J. Integer Seq. 10 Article 07.3.5, 1–21 (2007)

  5. A. Benyattou, M. Mihoubi, Curious congruences related to the Bell polynomials. Quaest. Math. 41(3), 437–448 (2018)

  6. F.R. Bernhart, Catalan, Motzkin, and Riordan numbers. Discrete Math. 204, 73–112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Björner, F. Brenti, Combinatorics of coxeter groups (Springer-Verlag, Berlin, 2005)

    MATH  Google Scholar 

  8. J. Blissard, Theory of generic equations. Quart. J. Pure Appl. Math. 4, 279–305 (1861)

    Google Scholar 

  9. J. Caicedo, V. Moll, J.L. Ramírez, D. Villamizar, Extensions of set partitions and permutations. Electron. J. Comb. 26, 1–45 (2019)

    MathSciNet  MATH  Google Scholar 

  10. W.Y.C. Chen, D.G.L. Wang, Singletons and adjacencies of set partitions of type \(B\). Discrete Math. 311, 418–422 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Dobiński, Summierung der Reihe \(\sum n^m/n!\) für \(m=1,2,\)\(3,4,5,\dots \). Arch. für Mat. und Physik. 61, 333–336 (1877)

    MATH  Google Scholar 

  12. N. Eriksen, R. Freij, and J. Wästlund. Enumeration of derangements with descents in prescribed positions. Electron. J. Combin. 16 (2009) #R32, 1–19

  13. G. Gordon, E. McMahon, Moving faces to other places: facet derangements. Am. Math. Monthly 117(10), 865–880 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Mező, Combinatorics and number theory of counting sequences (Chapman & Hall / CRC, Boca Raton, 2019)

    MATH  Google Scholar 

  15. I. Mező. The \(r\)-Bell numbers. J. Integer Seq. 14 Article 11.1.1 (2011), 1–14

  16. I. Mező, Á. Baricz, On the generalization of the Lambert \(W\) function. Trans. Am. Math. Soc. 369(11), 7917–7934 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. I. Mező, Code for the \(r\)-Lambert function on GitHUB: https://github.com/IstvanMezo/r-Lambert-function

  18. I. Mező, J.L. Ramírez, Divisibility properties of the \(r\)-Bell numbers and polynomials. J. Number Theory. 177, 136–152 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. I. Mező, J.L. Ramírez, A combinatorial approach to derangement matrix of type \(B\). Linear Algebra Appl. 582, 156–180 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. V.H. Moll, J.L. Ramírez, D. Villamizar, Combinatorial and arithmetical properties of the restricted and associated Bell and factorial numbers. J. Comb. 9, 693–720 (2018)

    MathSciNet  MATH  Google Scholar 

  21. J. Pitman, Some probabilistic aspects of set partitions. Am. Math. Monthly 104(3), 201–209 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Reiner, Non-crossing partitions for classical reflection groups. Discrete Math. 177, 195–222 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. G.C. Rota, B.D. Taylor, The classical umbral calculus. SIAM J. Math. Anal. 25, 694–711 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. L.W. Shapiro, S. Getu, W. Woan, L. Woodson, The Riordan group. Discrete Appl. Math. 34, 229–239 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. N.J.A. Sloane. The On-Line Encyclopedia of Integer Sequences, http://oeis.org

  26. M. Z. Spivey. A generalized recurrence for Bell numbers. J. Integer Seq. 11, Article 08.2.5, 1–3 (2008)

  27. Y. Sun, X. Wu, J. Zhuang, Congruences on the Bell polynomials and the derangement polynomials. J. Number Theory 133, 1564–1571 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Sun, X. Wu, The largest singletons of set partitions. Eur. J. Combin. 32, 369–382 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Y. Sun, Y. Xu, The largest singletons in weighted set partitions and its applications. Discrete Math. Theor. Comput. Sci. 13(3), 75–86 (2011)

    MathSciNet  MATH  Google Scholar 

  30. R. Suter, Two analogues of a classical sequence. J. Integer Seq. 3 Article 00.1.8. , 1–18 (2000)

  31. J. Touchard, Propriétés arithmétiques de certains nombres récurrents. Ann. Soc. Sci. Bruxelles A 53, 21–31 (1933)

    MATH  Google Scholar 

  32. H. Wilf, Generating Functionology, 2nd edn. (Academic Press, Cambridge, 1994)

    Google Scholar 

  33. T. Zaslavsky, The geometry of root systems and signed graphs. Am. Math. Monthly 88, 88–105 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referee for carefully reading the paper and giving helpful comments and suggestions. The first author thanks for hospitality the Department of Mathematics of Universidad Nacional de Colombia, Bogotá, Colombia, where the presented work was initiated. The second author was partially supported by Universidad Nacional de Colombia, Project No. 46240.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    István Mező

  2. John von Neumann Faculty of Informatics, Óbuda University, Bécsiút 96/b, Budapest, Hungary

    István Mező

  3. Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia

    José L. Ramírez

Authors
  1. István Mező
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  2. José L. Ramírez
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Correspondence to István Mező.

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Mező, I., Ramírez, J.L. Set partitions and partitions without singleton blocks of type B. Period Math Hung 85, 246–263 (2022). https://doi.org/10.1007/s10998-021-00439-1

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