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The Lucas sequence is a sequence of polynomials in s, and t definedrecursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. Onspecialization of s and t one can recover the Fibonacci numbers, thenonnegative integers, and the q-integers [n]_q. Given a quantity which isexpressed in terms of products and quotients of nonnegative integers, oneobtains 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 apolynomial in s and t with nonnegative integer coefficients and, if so, what itcounts. The first simple combinatorial interpretation for this polynomialanalogue of the binomial coefficients was given by Sagan and Savage, althoughtheir model resisted being used to prove identities for these Lucasnomials orextending their ideas to other combinatorial sequences. The purpose of thispaper is to give a new, even more natural model for these Lucasnomials usinglattice paths which can be used to prove various equalities as well asextending to Catalan numbers and their relatives, such as those for finiteCoxeter groups.

Combinatorial interpretations of Lucas analogues of binomial coefficients and Catalan numbers