A Generalization of the Catalan Numbers

Abstract

In this paper, we generalize the Catalan number $C_n$ to the $(m, n)$th Catalan number $C(m, n)$ using a combinatorial description, as follows: the number of paths in $\mathbb{R}^m$ from the origin to the point $(n,\cdots,n,(m-1)n) with $m$ kinds of moves such that the path never rises above the hyperplane $x_m=x_1+\cdots+x_{m-1}$.