A variant of Touchard's Catalan number identity

TL;DR

This paper explores a new combinatorial interpretation of Catalan numbers by counting polygon dissections based on specific triangle configurations, revealing a variant of Touchard's identity.

Contribution

It introduces a novel counting method for polygon dissections that leads to a new identity related to Touchard's Catalan number identity.

Findings

Derived a new combinatorial interpretation of Catalan numbers.

Established a connection between polygon dissections and Touchard's identity.

Provided a formula linking dissection counts to binomial coefficients.

Abstract

It is well known that the Catalan number C_n counts dissections of a regular (n+2)-gon into triangles. Here we count such dissections by number of triangles that contain two sides of the polygon among their three edges, leading to a combinatorial interpretation of the identity C_n =sum_{1<=k<=n/2} 2^{n-2k} n-choose-2k C_k (k(n+2))/(n(n-1)), and illustrating its connection with Touchard's identity.