Combinatorial interpretations of Lucas analogues of binomial coefficients and Catalan numbers

TL;DR

This paper introduces a new combinatorial lattice path model for Lucas analogues of binomial coefficients and Catalan numbers, enabling proofs of identities and extensions to related sequences.

Contribution

It provides a natural lattice path interpretation for Lucas analogues, facilitating combinatorial proofs and extensions to Catalan numbers and Coxeter group sequences.

Findings

Established a lattice path model for Lucas binomial coefficients

Proved identities using the new combinatorial interpretation

Extended the model to Catalan numbers and related sequences

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.