Mixed restricted Stirling numbers

  • S. Barati
  • B. BényiEmail author
  • A. Jafarzadeh
  • D. Yaqubi


We investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.

Key words and phrases

multiplicative partition function Stirling number of the second kind mixed partition of a set 

Mathematics Subject Classification

05A18 11B73 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors thank the anonymous referee for some useful comments.


  1. 1.
    M. Aigner, Combinatorial Theory, Springer Science and Business Media (2012).Google Scholar
  2. 2.
    Bóna, M., Mező, I.: Real zeros and partitions without singleton blocks. European J. Combin. 51, 500–510 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Broder, A.Z.: The \(r\)-Stirling numbers. Discrete Math. 49, 241–259 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carlitz, L.: Weighted Stirling numbers of the first and second kind - IT. Fibonacci Quart. 18, 242–257 (1980)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall CRC (2002)Google Scholar
  6. 6.
    Choi, J.Y., Smith, J.D.H.: On combinatorics of multi-restricted numbers. Ars Combin. 75, 45–63 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Choi, J.Y., Smith, J.D.H.: On the unimodality and combinatorics of Bessel numbers. Discrete Math. 264, 45–53 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co. (The Netherlands, 1974)Google Scholar
  9. 9.
    J. Culver and A.  Weingartner, Set partitions without blocks of certain sizes, arXiv: 1806.02316
  10. 10.
    P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press (Cambridge, 2009)Google Scholar
  11. 11.
    Howard, F.T.: Associated Stirling numbers. Fibonacci Quart. 18, 303–315 (1980)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Komatsu, T., Liptai, K., Mező, I.: Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers. Publ. Math. Debrecen. 88, 357–368 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    T. Mansour, Combinatorics of Set Partitions, CRC Press (2012)Google Scholar
  14. 14.
    Miksa, F.L., Moser, L., Wyman, M.: Restricted partitions of finite sets. Canad. Math. Bull. 1, 87–96 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    N. J. A. Sloane, The On-Line Encyclopaedia of Integer Sequences, (2018)Google Scholar
  16. 16.
    Wang, Ch., Mező, I.: Some limit theorems with respect to constrained permutations and partitions. Monatsh. Math. 182, 155–164 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yaqubi, D., Mirzavaziri, M., Saeednezhad, Y.: Mixed \(r\)-Stirling number of the second kind. Online J. Anal. Comb. 11, 5 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  • S. Barati
    • 1
  • B. Bényi
    • 2
    Email author
  • A. Jafarzadeh
    • 1
  • D. Yaqubi
    • 3
  1. 1.Department of Pure MathematicsFerdowsi University of MashhadMashhadIran
  2. 2.Faculty of Water SciencesNational University of Public ServiceBudapestHungary
  3. 3.Faculty of Agriculture and Animal ScienceUniversity of Torbat-e JamTorbat-e JamIran

Personalised recommendations