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Explicit formulae for generalized Stirling and Eulerian numbers (2404.16982v1)

Published 25 Apr 2024 in math.CO

Abstract: In this article we generalize the $q$-difference operator due to Carlitz in order to derive explicit sum formulae for several extensions of Stirling numbers of the second kind, including complete homogeneous symmetric functions, complementary symmetric functions, $r$-Whitney numbers and elliptic analogues of rook, Stirling and Lah numbers. Furthermore, we generalize Carlitz' $q$-Eulerian numbers to a Lagrange polynomial extension. We define them by generalizing Worpitzky's identity appropriately, and derive a recursion and an explicit sum formulae. Special cases include $r$-Whitney Eulerian numbers and elliptic Eulerian numbers.

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References (34)
  1. Elliptic and q𝑞qitalic_q-analogs of the Fibonomial numbers. SIGMA Symmetry Integrability Geom. Methods Appl., 16:Paper No. 076, 16 pp., 2020.
  2. Dan Betea. Elliptically distributed lozenge tilings of a hexagon. SIGMA Symmetry Integrability Geom. Methods Appl., 14:Paper No. 032, 39 pp., 2018.
  3. q𝑞qitalic_q-Distributions on boxed plane partitions. Selecta Math. (N.S.), 16(4):731–789, 2010.
  4. Francesco Brenti. q𝑞qitalic_q-Eulerian polynomials arising from Coxeter groups. European J. Combin., 15(5):417–441, 1994.
  5. Leonard Carlitz. On abelian fields. Trans. Amer. Math. Soc., 35(1):122–136, 1933.
  6. Leonard Carlitz. q𝑞qitalic_q-Bernoulli numbers and polynomials. Duke Math. J., 15:987–1000, 1948.
  7. Leonard Carlitz. q𝑞qitalic_q-Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc., 76:332–350, 1954.
  8. On the descent numbers and major indices for the hyperoctahedral group. Adv. in Appl. Math., 38(3):275–301, 2007.
  9. A Carlitz identity for the wreath product Cr≀𝔖n≀subscript𝐶𝑟subscript𝔖𝑛C_{r}\wr\mathfrak{S}_{n}italic_C start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ≀ fraktur_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Adv. in Appl. Math., 47(2):199–215, 2011.
  10. Johann Cigler. Operatormethoden für q𝑞qitalic_q-Identitäten. Monatsh. Math., 88(2):87–105, 1979.
  11. Explicit formulas for the first form (q,r)𝑞𝑟(q,r)( italic_q , italic_r )-Dowling numbers and (q,r)𝑞𝑟(q,r)( italic_q , italic_r )-Whitney–Lah numbers. Eur. J. Pure Appl. Math., 14(1):65–81, 2021.
  12. The complementary symmetric functions: connection constants using negative sets. Adv. Math., 135(2):207–219, 1998.
  13. Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2nd edition, 2004.
  14. Schur functions, Good’s identity, and hypergeometric series well poised in S⁢U⁢(n)𝑆𝑈𝑛{SU}(n)italic_S italic_U ( italic_n ). Adv. in Math., 48(2):177–188, 1983.
  15. Elliptic stirling numbers of the second and first kind. In preparation.
  16. Josef Küstner. Elliptic Combinatorics of Lattice Paths, Domino Tilings and Rook Placements. University of Vienna, 2022. PhD Thesis.
  17. Lattice paths and negatively indexed weight-dependent binomial coefficients. Annals of Combinatorics, 27:1–39, 2023.
  18. Some combinatorial identities of the r𝑟ritalic_r-Whitney-Eulerian numbers. Appl. Anal. Discrete Math., 13(2):378–398, 2019.
  19. István Mező. A new formula for the Bernoulli polynomials. Results Math., 58(3-4):329–335, 2010.
  20. Louis M. Milne-Thomson. The Calculus Of Finite Differences. Macmillan And Company., Limited, 1933. PhD Thesis.
  21. T. Kyle Petersen. Eulerian numbers. Birkhäuser/Springer, New York, 2015. With a foreword by Richard Stanley.
  22. Eulerian numbers associated with arithmetical progressions. Electron. J. Combin., 25(1):Paper No. 1.48, 12 pp., 2018.
  23. Rook theory, generalized Stirling numbers and (p,q)𝑝𝑞(p,q)( italic_p , italic_q )-analogues. Electron. J. Combin., 11(1):Research Paper 84, 48 pp., 2004.
  24. Hjalmar Rosengren. Elliptic hypergeometric functions. In Howard S. Cohl and Mourad E. H. Ismail, editors, Lectures on Orthogonal Polynomials and Special Functions, pages 213–279. Cambridge University Press, Cambridge, 2020.
  25. Gian-Carlo Rota, editor. Finite operator calculus. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. With the collaboration of P. Doubilet, C. Greene, D. Kahaner, A. Odlyzko and R. Stanley.
  26. q𝑞qitalic_q-Stirling numbers in type B. European Journal of Combinatorics, 118:103899, 2024.
  27. Michael J. Schlosser. Elliptic enumeration of nonintersecting lattice paths. J. Combin. Theory Ser. A, 114(3):505–521, 2007.
  28. Michael J. Schlosser. A noncommutative weight-dependent generalization of the binomial theorem. Sém. Lothar. Combin., 81:Art. B81j, 24 pp., 2020.
  29. Michael J. Schlosser. An elliptic extension of the multinomial theorem. Preprint, arXiv:2307.12921, 2023.
  30. Log-concavity results for a biparametric and an elliptic extension of the q𝑞qitalic_q-binomial coefficients. Int. J. Number Theory, 17(3):787–804, 2021.
  31. Basic hypergeometric summations from rook theory. In Analytic number theory, modular forms and q𝑞qitalic_q-hypergeometric series, volume 221 of Springer Proc. Math. Stat., pages 677–692. Springer, Cham, 2017.
  32. Elliptic rook and file numbers. Electron. J. Combin., 24(1):Paper No. 1.31, 47 pp., 2017.
  33. Richard P. Stanley. Enumerative Combinatorics 2. Cambridge Univ. Press, Cambridge, 2001.
  34. Heinrich Weber. Elliptische Functionen und algebraische Zahlen. Vieweg-Verlag, Braunschweig, 1891.

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