A combinatorial proof of a formula for the Lucas-Narayana polynomials |
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Affiliation: | 1. St. Olaf College, 1520 St. Olaf Avenue, Northfield, 55057, MN, USA;2. Pepperdine University, 24255 Pacific Coast Highway, Malibu, 90263, CA, USA |
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Abstract: | In 2020, Bennett, Carrillo, Machacek and Sagan gave a polynomial generalization of the Narayana numbers and conjectured that these polynomials have positive integer coefficients for and for . In 2020, Sagan and Tirrell used a powerful algebraic method to prove this conjecture (in fact, they extend and prove the conjecture for more than just the type A case). In this paper we give a combinatorial proof of a formula satisfied by the Lucas-Narayana polynomials described by Bennett et al. This gives a combinatorial proof that these polynomials have positive integer coefficients. A corollary of our main result establishes a parallel theorem for the FiboNarayana numbers , providing a combinatorial proof of the conjecture that these are positive integers for . |
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Keywords: | Narayana number Fibonomial coefficient Lucasnomial Lucas-Narayana polynomial |
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