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


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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