Super FiboCatalan Numbers and their Lucas Analogues

TL;DR

This paper introduces super FiboCatalan numbers and their Lucas analogues, proving their integrality and exploring their properties through combinatorial and algebraic methods.

Contribution

It defines new FiboCatalan variants and Lucas analogues, establishing their polynomial nature and integrality, extending the understanding of super Catalan numbers.

Findings

Super FiboCatalan numbers are integers.

Lucas analogues are polynomials with non-negative coefficients.

New combinatorial interpretations are explored.

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.