Combinatorial Interpretations of Lucas Analogues of Binomial Coefficients and Catalan Numbers

Abstract

The Lucas sequence is a sequence of polynomials in st defined recursively by \(\{0\}=0\), \(\{1\}=1\), and \(\{n\}=s\{n-1\}+t\{n-2\}\) for \(n\ge 2\). 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 st 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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Acknowledgements

We had helpful discussions with Nantel Bergeron, Cesar Ceballos, and Volker Strehl about this work.

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Correspondence to Bruce E. Sagan.

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Bennett, C., Carrillo, J., Machacek, J. et al. Combinatorial Interpretations of Lucas Analogues of Binomial Coefficients and Catalan Numbers. Ann. Comb. (2020). https://doi.org/10.1007/s00026-020-00500-9

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Keywords

  • Binomial coefficient
  • Catalan number
  • Combinatorial interpretation
  • Coxeter group
  • Generating function
  • Integer partition
  • Lattice path
  • Lucas sequence
  • Tiling

Mathematics Subject Classification

  • Primary 05A10
  • Secondary 05A15
  • 05A19
  • 11B39