Super-Catalan Numbers of the Third and Fourth Kind

TL;DR

This paper investigates the combinatorial interpretations of Super-Catalan numbers for specific cases, providing solutions for m=3,4, and demonstrating the absence of solutions for m=5.

Contribution

It offers explicit combinatorial interpretations for Super-Catalan numbers when m=3,4, advancing understanding of these sequences and highlighting limitations for m=5.

Findings

Explicit combinatorial interpretations found for m=3,4

No combinatorial interpretation exists for m=5

Advances understanding of Super-Catalan numbers' combinatorial nature

Abstract

The Super-Catalan numbers are a generalization of the Catalan numbers defined as $T(m,n) = \frac{(2m)!(2n)!}{2m!n!(m+n)!}$. It is an open problem to find a combinatorial interpretation for $T(m,n)$. We resolve this for $m=3,4$ using a common form; no such solution exists for $m=5$.