Some applications of S-restricted set partitions


In the paper, the authors present several new relations and applications for the combinatorial sequence that counts the possible partitions of a finite set with the restriction that the size of each block is contained in a given set. One of the main applications is in the study of lonesum matrices.

This is a preview of subscription content, log in to check access.


  1. 1.

    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)

  2. 2.

    H. Belbachir and I. E. Bousbaa, Associated Lah numbers and \(r\)-Stirling numbers, 1–23 (2014), arXiv:1404.5573v2

  3. 3.

    B. Bényi, Advances in Bijective Combinatorics, Ph.D thesis (2014). benyi-beata-d.pdf

  4. 4.

    B. Bényi, P. Hajnal, Combinatorics of poly-Bernoulli numbers. Stud. Sci. Math. Hung. 52(4), 537–558 (2015)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    B. Benyi, P. Hajnal, Combinatorial properties of poly-Bernoulli relatives. Integers 17, A31 (2017)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    B. Bényi, Restricted lonesum matrices, 1–12 (2017), arXiv:1711.10178v2

  7. 7.

    M. Bóna, I. Mező, Real zeros and partitions without singleton blocks. Eur. J. Comb. 51, 500–510 (2016)

  8. 8.

    C. Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues. Integers 8, A02 (2008)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    A.Z. Broder, The \(r\)-Stirling numbers. Discrete Math. 49, 241–259 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    G.-S. Cheon, J.-H. Jung, \(r\)-Whitney numbers of Dowling lattices. Discrete Math. 312, 2337–2348 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    J.Y. Choi, J.D.H. Smith, On the unimodality and combinatorics of Bessel numbers. Discrete Math. 264, 45–53 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    J.Y. Choi, J.D.H. Smith, Reciprocity for multi-restricted numbers. J. Comb. Theory Ser. A. 113, 1050–1060 (2006)

    Article  MATH  Google Scholar 

  13. 13.

    L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions (D. Reidel Publishing Co., Dordrecht, 1974)

    Google Scholar 

  14. 14.

    T. Diagana, H. Maïga, Some new identities and congruences for Fubini numbers. J. Number. Theory 173, 547–569 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

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

    MATH  Google Scholar 

  16. 16.

    J. Engbers, D. Galvin, and C. Smyth, Restricted Stirling and Lah numbers and their inverses. J. Combin. Theory Ser. A. 161, 271–298 (2016)

  17. 17.

    P. Flajolet, R. Sedgewick, Analytic Combinatorics (Cambridge University Press, Cambridge, 2009)

    Google Scholar 

  18. 18.

    Z. Füredi, P. Hajnal, Davenport–Schinzel theory of matrices. Discrete Math. 103, 233–251 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    K. Kamano, Lonesum decomposable matrices. Discrete Math. 341(2), 341–349 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    M. Kaneko, Poly-Bernoulli numbers. J. Théor. Nombres Bordx 9, 199–206 (1997)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    T. Komatsu, Generalized incomplete poly-Bernoulli and poly-Cauchy numbers. Period. Math. Hung. 75(1), 96–113 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    T. Komatsu, K. Liptai, I. Mező, Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers. Publ. Math. Debr. 88, 357–368 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    T. Komatsu, J. L. Ramírez, Some determinants involving incomplete Fubini numbers. An. Şt. Univ. Ovidius Constanţa, 1–28 (2018), arXiv:1802.06188 (to appear)

  24. 24.

    T. Komatsu, J.L. Ramírez, Generalized poly-Cauchy and poly-Bernoulli numbers by using incomplete \(r\)-Stirling numbers. Aequ. Math. 91, 1055–1071 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    T. Mansour, M. Schork, Commutations Relations, Normal Ordering, and Stirling numbers (CRC Press, Boca Roton, 2015)

    Google Scholar 

  26. 26.

    T. Mansour, M. Shattuck, A polynomial generalization of some associated sequences related to set partitions. Period. Math. Hung. 75(2), 398–412 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    I. Mező, J.H. Jung, J.L. Ramírez, The \(r\)-Bessel and restricted \(r\)-Bell numbers. Aust. J. Comb. 70(2), 202–220 (2018)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    V. Moll, J. L. Ramírez, D. Villamizar, Combinatorial and arithmetical properties of the restricted and associated Bell and factorial numbers. J. Comb. 1–20 (2017), arXiv:1707.08138 (to appear)

  29. 29.

    S. Nkonkobe, V. Murali, On some properties and relations between restricted barred preferential arrangements, multi-poly-Bernoulli numbers and related numbers, 1–12 (2015), arXiv:1509.07352

  30. 30.

    J. Pippenger, The hypercube of resistors, asymptotic expansions, and the preferential arrangements. Math. Mag. 83(5), 331–346 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

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

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    B. Poonen, Periodicity of a combinatorial sequence. Fibonacci Quart. 26(1), 70–76 (1988)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    H.J. Ryser, Combinatorial properties of matrices of zeros and ones. Can. J. Math. 9, 371–377 (1957)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

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

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    N. J. A. Sloane, The on-line encyclopedia of integer sequences. Accessed 2018

  36. 36.

    T. Wakhare, Refinements of the Bell and Stirling numbers, Trans. Comb. 1–17 (2018), arXiv:1710.02956 (to appear)

Download references


The authors would like to thank the anonymous referee for some useful comments. The research of José L. Ramírez was partially supported by Universidad Nacional de Colombia, Project No. 37805.

Author information



Corresponding author

Correspondence to José L. Ramírez.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bényi, B., Ramírez, J.L. Some applications of S-restricted set partitions. Period Math Hung 78, 110–127 (2019).

Download citation


  • Combinatorial identities
  • Generating functions
  • S-restricted Stirling numbers
  • Lonesum matrices
  • Poly-Bernoulli numbers

Mathematics Subject Classification

  • Primary 11B83
  • Secondary 11B73
  • 05A19
  • 05A15