2023/08/25 by Kendra Killpatrick, Killpatrick, Kendra
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2308.13457
openalex publication_date 2023/08/25 · openalex created_date 2023/08/30 · openalex updated_date 2026/07/28
Catalan observed in 1874 that the numbers S(m,n) = ((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 ≤ s ≤ 4. In this paper, we define the super FiboCatalan numbers S(m,n)F = \fracF2m! F2n!Fm! Fn! Fm+n! and the generalized FiboCatalan numbers Jr,F \fracF2n!Fn! Fn+r+1! where Jr,F = \fracF2r+1!Fr!. 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.