Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences

Details Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences

2009-11-16

arxiv.org/abs/0911.3159v1

Mathematics

Combinatorics

7 pages, 2 figures

Scientific paper

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 corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by C{n,k}={n}!/({k}!{n-k}!) where {n}!={1}{2}...{n}. It is easy to see that C{n,k} is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k by n-k rectangle. When s=t=1 we obtain combinatorial interpretations of the fibonomial coefficients which are simpler than any that have previously appeared in the literature.

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