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A combinatorial proof of a formula for the Lucas-Narayana polynomials.
- Source :
-
Discrete Mathematics . Dec2022, Vol. 345 Issue 12, pN.PAG-N.PAG. 1p. - Publication Year :
- 2022
-
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 1 ≤ k ≤ n and for n ≥ 1. 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 N n , k , F , providing a combinatorial proof of the conjecture that these are positive integers for n ≥ 1. [ABSTRACT FROM AUTHOR]
- Subjects :
- *INTEGERS
*LOGICAL prediction
*GENERALIZATION
Subjects
Details
- Language :
- English
- ISSN :
- 0012365X
- Volume :
- 345
- Issue :
- 12
- Database :
- Academic Search Index
- Journal :
- Discrete Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 159289768
- Full Text :
- https://doi.org/10.1016/j.disc.2022.113077