The Lucas sequence is a sequence of polynomials in s, t 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 s, 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.

Lucas序列是s中的多项式序列，  t由\（\ {0 \} = 0 \），\（\ {1 \} = 1 \）和\（\ {n \} = s \ {n-1 \} + t \ {n-2 \} \）表示\（n \ ge 2 \）。上的专业化小号和吨一个可以收回斐波那契数，所述非负整数，而q -integers \（[N] _q \） 。给定一个以非负整数的乘积和商表示的量，可以通过用\（\ {n \} \）替换表达式中的n个因子来获得Lucas类似物。。然后自然会问得出的有理函数是否实际上是s，  t中具有非负整数系数的多项式，如果是，则计算出什么。Sagan和Savage对此二项式系数的多项式类似物进行了第一个简单的组合解释，尽管他们的模型拒绝用来证明这些Lucasnomial的身份或将其思想扩展到其他组合序列。本文的目的是使用晶格路径为这些Lucasnomial提供一个新的甚至更自然的模型，该模型可用于证明各种相等性以及扩展到加泰罗尼亚数及其亲属，例如有限的Coxeter群。