Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.8

A Generalization of the Catalan Numbers


Reza Kahkeshani
Department of Pure Mathematics
Faculty of Mathematical Sciences
University of Kashan
Kashan
Iran

Abstract:

In this paper, we generalize the Catalan number Cn 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 $\bigl( \underbrace{n,\ldots,n}_{m-1},(m-1)n \bigr)$ with m kinds of moves such that the path never rises above the hyperplane $x_m=x_1+\cdots+x_{m-1}$.

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(Concerned with sequence A000108.)

Received March 16 2013; revised version received July 3 2013. Published in Journal of Integer Sequences, July 30 2013.

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