Multi-Dimensional Catalan Numbers

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 Tmn(,); finding and proving the relation between Cn[a] and Tmn(,) and proving the formulae on Tmn(,). Finally, this work solves a variant of the famous Ballot Problem by using the Hook length formula, pointing for future directions.