Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 076, 16 pages      arXiv:1911.12785      https://doi.org/10.3842/SIGMA.2020.076
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

Elliptic and $q$-Analogs of the Fibonomial Numbers

Nantel Bergeron a, Cesar Ceballos b and Josef Küstner c
a) Department of Mathematics and Statistics, York University, Toronto, Canada
b) Institute of Geometry, TU Graz, Graz, Austria
c) Faculty of Mathematics, University of Vienna, Vienna, Austria

Received March 14, 2020, in final form July 29, 2020; Published online August 13, 2020

Abstract
In 2009, Sagan and Savage introduced a combinatorial model for the Fibonomial numbers, integer numbers that are obtained from the binomial coefficients by replacing each term by its corresponding Fibonacci number. In this paper, we present a combinatorial description for the $q$-analog and elliptic analog of the Fibonomial numbers. This is achieved by introducing some $q$-weights and elliptic weights to a slight modification of the combinatorial model of Sagan and Savage.

Key words: Fibonomial; Fibonacci; $q$-analog; elliptic analog; weighted enumeration.

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References

  1. Amdeberhan T., Chen X., Moll V.H., Sagan B.E., Generalized Fibonacci polynomials and Fibonomial coefficients, Ann. Comb. 18 (2014), 541-562, arXiv:1306.6511.
  2. Athanasiadis C.A., On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005), 179-196.
  3. Aval J.C., Bergeron F., A note on: rectangular Schröder parking functions combinatorics, Sém. Lothar. Combin. 79 (2018), Art. B79a, 13 pages, arXiv:1603.09487.
  4. Benjamin A.T., Reiland E., Combinatorial proofs of Fibonomial identities, Fibonacci Quart. 52 (2014), 28-34.
  5. Bennett C., Carrillo J., Machacek J., Sagan B.E., Combinatorial interpretations of Lucas analogues of binomial coefficients and Catalan numbers, arXiv:1809.09036.
  6. Bergeron F., Open questions for operators related to rectangular Catalan combinatorics, J. Comb. 8 (2017), 673-703, arXiv:1603.04476.
  7. Bergeron F., Garsia A., Leven E.S., Xin G., Some remarkable new plethystic operators in the theory of Macdonald polynomials, J. Comb. 7 (2016), 671-714, arXiv:1405.0316.
  8. Bergeron F., Garsia A., Sergel Leven E., Xin G., Compositional $(km,kn)$-shuffle conjectures, Int. Math. Res. Not. 2016 (2016), 4229-4270, arXiv:1404.4616.
  9. Borodin A., Gorin V., Rains E.M., $q$-distributions on boxed plane partitions, Selecta Math. (N.S.) 16 (2010), 731-789, arXiv:0905.0679.
  10. Chen X., Sagan B.E., The fractal nature of the Fibonomial triangle, Integers 14 (2014), A3, 12 pages, arXiv:1306.2377.
  11. Cherednik I., Double affine Hecke algebras, London Mathematical Society Lecture Note Series, Vol. 319, Cambridge University Press, Cambridge, 2005.
  12. Dunlap R.A., The golden ratio and Fibonacci numbers, World Sci. Publ. Co., Inc., River Edge, NJ, 1997.
  13. Etingof P., Ginzburg V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243-348, arXiv:math.AG/0011114.
  14. Etingof P., Gorsky E., Losev I., Representations of rational Cherednik algebras with minimal support and torus knots, Adv. Math. 277 (2015), 124-180, arXiv:1304.3412.
  15. Etingof P., Ma X., Lecture notes on Cherednik algebras, arXiv:1001.0432.
  16. Gorsky E., Mazin M., Vazirani M., Affine permutations and rational slope parking functions, Trans. Amer. Math. Soc. 368 (2016), 8403-8445, arXiv:1403.0303.
  17. Gorsky E., Neguţ A., Refined knot invariants and Hilbert schemes, J. Math. Pures Appl. 104 (2015), 403-435, arXiv:1304.3328.
  18. Gorsky E., Oblomkov A., Rasmussen J., Shende V., Torus knots and the rational DAHA, Duke Math. J. 163 (2014), 2709-2794, arXiv:1207.4523.
  19. Hikita T., Affine Springer fibers of type $A$ and combinatorics of diagonal coinvariants, Adv. Math. 263 (2014), 88-122, arXiv:1203.5878.
  20. Hoggatt Jr. V.E., Long C.T., Divisibility properties of generalized Fibonacci polynomials, Fibonacci Quart. 12 (1974), 113-120.
  21. Li S.X., Personal communication, 2015, Algebraic Combinatorics Seminar at the Fields Institute.
  22. Mellit A., Toric braids and $(m,n)$-parking functions, arXiv:1604.07456.
  23. Posamentier A.S., Lehmann I., The (fabulous) Fibonacci numbers, Prometheus Books, Amherst, NY, 2007.
  24. Rains E.M., $BC_n$-symmetric Abelian functions, Duke Math. J. 135 (2006), 99-180, arXiv:math.CO/0402113.
  25. Reiland E., Combinatorial Interpretations of Fibonomial identities, Master's Thesis, Harvey Mudd College, Claremont, CA, 2011.
  26. Rosengren H., Elliptic hypergeometric functions, Lecture notes at OPSF-S6, College Park, Maryland, August 2016, arXiv:1608.06161.
  27. Sagan B.E., Savage C.D., Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences, Integers 10 (2010), A52, 697-703, arXiv:0911.3159.
  28. Schlosser M.J., Elliptic enumeration of nonintersecting lattice paths, J. Combin. Theory Ser. A 114 (2007), 505-521, arXiv:math.CO/0602260.
  29. Schlosser M.J., A noncommutative weight-dependent generalization of the binomial theorem, Sém. Lothar. Combin. 81 (2020), Art. B81j, 20 pages, arXiv:1106.2112.
  30. Schlosser M.J., Senapati K., Uncu A.K., Log-concavity results for a biparametric and an elliptic extension of the $q$-binomial coefficients, arXiv:2002.07796.
  31. Schlosser M.J., Yoo M., Elliptic rook and file numbers, Electron. J. Combin. 24 (2017), 1.31, 47 pages, arXiv:1512.01720.
  32. Schlosser M.J., Yoo M., Weight-dependent commutation relations and combinatorial identities, Discrete Math. 341 (2018), 2308-2325, arXiv:1610.08680.
  33. Thiel M., From Anderson to zeta, Adv. in Appl. Math. 81 (2016), 156-201, arXiv:1504.07363.
  34. Tirrell J., Sagan B.E., Lucas atoms, arXiv:1909.02593.
  35. Weber H., Elliptische functionen und algebraische zahlen, Vieweg-Verlag, Braunschweig, 1891.

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