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  • combinatorial interpretations of Binomial Coefficient analogues related to lucas sequences
    Integers, 2010
    Co-Authors: Bruce E Sagan, Carla D Savage
    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 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 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx 2

  • combinatorial interpretations of Binomial Coefficient analogues related to lucas sequences
    arXiv: Combinatorics, 2009
    Co-Authors: Bruce E Sagan, Carla D Savage
    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 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.