Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences

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 {n k } = {n}! {k}! {n− k}! where {n}! = {1} {2} · · · {n}. It is easy to see that { 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 × (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. ∗Work partially done while a Program Officer at NSF. The views expressed are not necessarily those of the NSF. †Partially supported by NSA grant H98230-08-1-0072