Super FiboCatalan Numbers and their Lucas Analogues

Abstract

Catalan observed in 1874 that the numbers $S(m,n) = \frac{(2m)! (2n)!}{m! n! (m+n)!}$, now called the super Catalan numbers, are integers but there is still no known combinatorial interpretation for them in general, although interpretations have been given for the case $m=2$ and for $S(m, m+s)$ for $0 \leq s \leq 4$. In this paper, we define the super FiboCatalan numbers $S(m,n)_F = \frac{F_{2m}! F_{2n}!}{F_m! F_n! F_{m+n}!}$ and the generalized FiboCatalan numbers $J_{r,F} \frac{F_{2n}!}{F_n! F_{n+r+1}!}$ where $J_{r,F} = \frac{F_{2r+1}!}{F_r!}$. In addition, we give Lucas analogues for both of these numbers and use a result of Sagan and Tirrell to prove that the Lucas analogues are polynomials with non-negative integer coefficients which in turn proves that the super FiboCatalan numbers and the generalized FiboCatalan numbers are integers.