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      Combinatorial interpretations of Lucas analogues of binomial coefficients and Catalan numbers

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          Abstract

          The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers [n]_q. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of n in the expression with {n}. It is then natural to ask if the resulting rational function is actually a polynomial in s and t with nonnegative integer coefficients and, if so, what it counts. The first simple combinatorial interpretation for this polynomial analogue of the binomial coefficients was given by Sagan and Savage, although their model resisted being used to prove identities for these Lucasnomials or extending their ideas to other combinatorial sequences. The purpose of this paper is to give a new, even more natural model for these Lucasnomials using lattice paths which can be used to prove various equalities as well as extending to Catalan numbers and their relatives, such as those for finite Coxeter groups.

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          Author and article information

          Journal
          24 September 2018
          Article
          1809.09036
          c0bf1e0e-3ebd-47b2-b119-41d19d149ebc

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          05A10 (Primary) 05A15, 05A19, 11B39 (Secondary)
          26 pages
          math.CO

          Combinatorics
          Combinatorics

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