Multi-Dimensional Catalan Numbers
DOI:
https://doi.org/10.61173/0xnhka79Keywords:
Catalan numbers, Hook length formula, Ber-trand’s ballot problem, Dyck pathsAbstract
This article reviews the basics of the Catalan numbers and introduces a geometric proof for its formulae. Then, the work generalizes the Catalan numbers towards multi-dimensions. This paper later proposes the crucial steps in the foundation of the formulae on Cn [a] using Dyck paths, including defining Cn [a]; introducing the concept of T mn (,); finding and proving the relation between Cn [a] and T mn (,) and proving the formulae on T mn (,). Finally, this work solves a variant of the famous Ballot Problem by using the Hook length formula, pointing for future directions.
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