A Combinatorial Interpretation of The Numbers $6(2n)!/n!(n+2)!$

Abstract

It is well known that the numbers $(2m)!(2n)!/m!n!(m+n)!$ are integers, but in general there is no known combinatorial interpretation for them. When $m=0$ these numbers are the middle binomial coefficients $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$, and when $m=1$ they are twice the Catalan numbers. In this paper, we give combinatorial interpretations for these numbers when $m=2$ or 3.

11 pages, 2 figures