Periodica Mathematica Hungarica

, Volume 78, Issue 1, pp 110–127 | Cite as

Some applications of S-restricted set partitions

  • Beáta Bényi
  • José L. RamírezEmail author


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.


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

Mathematics Subject Classification

Primary 11B83 Secondary 11B73 05A19 05A15 



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.


  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)Google Scholar
  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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    B. Benyi, P. Hajnal, Combinatorial properties of poly-Bernoulli relatives. Integers 17, A31 (2017)MathSciNetzbMATHGoogle 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)Google Scholar
  8. 8.
    C. Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues. Integers 8, A02 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    A.Z. Broder, The \(r\)-Stirling numbers. Discrete Math. 49, 241–259 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    G.-S. Cheon, J.-H. Jung, \(r\)-Whitney numbers of Dowling lattices. Discrete Math. 312, 2337–2348 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J.Y. Choi, J.D.H. Smith, On the unimodality and combinatorics of Bessel numbers. Discrete Math. 264, 45–53 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J.Y. Choi, J.D.H. Smith, Reciprocity for multi-restricted numbers. J. Comb. Theory Ser. A. 113, 1050–1060 (2006)CrossRefzbMATHGoogle Scholar
  13. 13.
    L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions (D. Reidel Publishing Co., Dordrecht, 1974)zbMATHGoogle Scholar
  14. 14.
    T. Diagana, H. Maïga, Some new identities and congruences for Fubini numbers. J. Number. Theory 173, 547–569 (2017)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle 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)Google Scholar
  17. 17.
    P. Flajolet, R. Sedgewick, Analytic Combinatorics (Cambridge University Press, Cambridge, 2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Z. Füredi, P. Hajnal, Davenport–Schinzel theory of matrices. Discrete Math. 103, 233–251 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    K. Kamano, Lonesum decomposable matrices. Discrete Math. 341(2), 341–349 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. Kaneko, Poly-Bernoulli numbers. J. Théor. Nombres Bordx 9, 199–206 (1997)MathSciNetzbMATHGoogle Scholar
  21. 21.
    T. Komatsu, Generalized incomplete poly-Bernoulli and poly-Cauchy numbers. Period. Math. Hung. 75(1), 96–113 (2017)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    T. Mansour, M. Schork, Commutations Relations, Normal Ordering, and Stirling numbers (CRC Press, Boca Roton, 2015)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    J. Pitman, Some probabilistic aspects of set partitions. Am. Math. Mon. 104(3), 201–209 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    B. Poonen, Periodicity of a combinatorial sequence. Fibonacci Quart. 26(1), 70–76 (1988)MathSciNetzbMATHGoogle Scholar
  33. 33.
    H.J. Ryser, Combinatorial properties of matrices of zeros and ones. Can. J. Math. 9, 371–377 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    L.W. Shapiro, S. Getu, W. Woan, L. Woodson, The Riordan group. Discrete Appl. Math. 34, 229–239 (1991)MathSciNetCrossRefzbMATHGoogle 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)

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Faculty of Water SciencesNational University of Public ServiceBudapestHungary
  2. 2.Departamento de MatemáticasUniversidad Nacional de ColombiaBogotáColombia

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