A proof of Catalan's Convolution formula

TL;DR

This paper presents a new combinatorial proof of Catalan's convolution formula using polygonal dissections and also derives a formula for the average number of cycles in a triangulation.

Contribution

It introduces a novel proof technique for Catalan's convolution formula based on polygonal dissections and provides a new formula for average cycles in triangulations.

Findings

New proof of Catalan's convolution formula

Formula for average number of cycles in triangulations

Enumeration method using polygonal dissections

Abstract

We give a new proof of the $k$-fold convolution of the Catalan numbers. This is done by enumerating a certain class of polygonal dissections called $k$-in-$n$ dissections. Furthermore, we give a formula for the average number of cycles in a triangulation.