# Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences - Mathematics > Combinatorics

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Abstract: Let s and t be variables. Define polynomials {n} in s, t by {0}=0, {1}=1, and{n}=s{n-1}+t{n-2} for n >= 2. If s, t are integers then the correspondingsequence of integers is called a Lucas sequence. Define an analogue of thebinomial coefficients byC{n,k}={n}!-{k}!{n-k}! where {n}!={1}{2}

.{n}. It is easy to see thatC{n,k} is a polynomial in s and t. The purpose of this note is to give twocombinatorial interpretations for this polynomial in terms of statistics oninteger partitions inside a k by n-k rectangle. When s=t=1 we obtaincombinatorial interpretations of the fibonomial coefficients which are simplerthan any that have previously appeared in the literature.

Autor: Bruce Sagan Mathematics, Michigan State University, Carla Savage Computer Science, North Carolina State University

Fuente: https://arxiv.org/