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A generalized class of restricted Stirling and Lah numbers

  • Toufik Mansour and Mark Shattuck EMAIL logo
Published/Copyright: August 6, 2018
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 68 Issue 4

Abstract

In this paper, we consider a polynomial generalization, denoted by uma,b (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then uma,b (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by uma,b (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for uma,b (n, k) as well as for the associated Cauchy numbers.

MSC 2010: Primary 11B73; Secondary 05A19; 05A18
Keywords: restricted Stirling numbers; Cauchy numbers; polynomial generalization; combinatorial identities

  1. Communicated by Anatolij Dvurečenskij

References

[1] Choi, J. Y.: Multi-restrained Stirling numbers, Ars Combin. 120 (2015), 113–127.Search in Google Scholar

[2] Choi, J. Y.—Long, L.—Ng, S.—H.—Smith, J.: Reciprocity for multirestricted numbers, J. Combin. Theory Ser. A 113 (2006), 1050–1060.10.1016/j.jcta.2005.10.001Search in Google Scholar

[3] Graham, R. L.—Knuth, D. E.—Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, second edition, Addison-Wesley, Boston, 1994.Search in Google Scholar

[4] Komatsu, T.: Incomplete poly-Cauchy numbers, Monatsh. Math. 180 (2016), 271–288.10.1007/s00605-015-0810-zSearch in Google Scholar

[5] Komatsu, T.—Liptai, K.—Mezo, I.: Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers, (2015). arXiv:1510.05799v2.Search in Google Scholar

[6] Komatsu, T.—Mező, I.—Szalay, L.: Incomplete Cauchy numbers, Acta Math. Hungar. 149 (2016), 306–323.10.1007/s10474-016-0616-zSearch in Google Scholar

[7] Lah, I.: Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, Mitteil.-Bl. Math. Statistik 7 (1955), 203–212.Search in Google Scholar

[8] Mansour, T.—Schork, M.—Shattuck, M.: On a new family of generalized Stirling and Bell numbers, Electron. J. Combin. 18 (2011), #P77.10.37236/564Search in Google Scholar

[9] Mező, I.: Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17 (2014), Art. 14.1.1.Search in Google Scholar

[10] Sloane, N. J.: The On-Line Encyclopedia of Integer Sequences, 2010. http://oeis.org.Search in Google Scholar

Received: 2016-09-04
Accepted: 2017-04-11
Published Online: 2018-08-06
Published in Print: 2018-08-28

© 2018 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. A multi-parameter generalization of the symmetric algorithm
  2. Single identities forcing lattices to be Boolean
  3. Weighted uniform density ideals
  4. A generalized class of restricted Stirling and Lah numbers
  5. The Riemann hypothesis and universality of the Riemann zeta-function
  6. Distance functions on the sets of ordinary elliptic curves in short Weierstrass form over finite fields of characteristic three
  7. The drazin inverse of the sum of two matrices
  8. Refinements of the majorization-type inequalities via green and fink identities and related results
  9. Characterizations and properties of graphs of Baire functions
  10. Some improvements of the young mean inequality and its reverse
  11. On characteristic of bounded analytic functions involving hyperbolic derivative
  12. A uniqueness problem for entire functions related to Brück’s conjecture
  13. System of nonlocal resonant boundary value problems involving p-Laplacian
  14. Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients
  15. Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces
  16. Characterization of rough weighted statistical limit set
  17. Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers
  18. On generalized 4-th root metrics of isotropic scalar curvature
  19. Compactifications from generators and relations
  20. Diophantine quadruples with values in k-generalized Fibonacci numbers
https://doi.org/10.1515/ms-2017-0140
Keywords for this article
restricted Stirling numbers; Cauchy numbers; polynomial generalization; combinatorial identities

Articles in the same Issue

  1. A multi-parameter generalization of the symmetric algorithm
  2. Single identities forcing lattices to be Boolean
  3. Weighted uniform density ideals
  4. A generalized class of restricted Stirling and Lah numbers
  5. The Riemann hypothesis and universality of the Riemann zeta-function
  6. Distance functions on the sets of ordinary elliptic curves in short Weierstrass form over finite fields of characteristic three
  7. The drazin inverse of the sum of two matrices
  8. Refinements of the majorization-type inequalities via green and fink identities and related results
  9. Characterizations and properties of graphs of Baire functions
  10. Some improvements of the young mean inequality and its reverse
  11. On characteristic of bounded analytic functions involving hyperbolic derivative
  12. A uniqueness problem for entire functions related to Brück’s conjecture
  13. System of nonlocal resonant boundary value problems involving p-Laplacian
  14. Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients
  15. Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces
  16. Characterization of rough weighted statistical limit set
  17. Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers
  18. On generalized 4-th root metrics of isotropic scalar curvature
  19. Compactifications from generators and relations
  20. Diophantine quadruples with values in k-generalized Fibonacci numbers
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