2009/11/16 by Bruce Sagan, Sagan, Bruce, Carla Savage +1
Mathematics · #05A10 #05A17 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A10 #msc:05A17
paper · pdf · doi:10.48550/arxiv.0911.3159
7 pages, 2 figures
arxiv created 2009/11/16 · arxiv updated 2009/12/01
Let s and t be variables. Define polynomials n in s, t by 0=0, 1=1, and n=sn-1+tn-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 Cn,k=n!/(k!n-k!) where n!=12...n. It is easy to see that Cn,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.