Ears of triangulations and Catalan numbers

Abstract

It is known that a convex polygon of n sides admits C n-2 triangulations, where C n is a Catalan number. We classify these triangulations (considered as outerplanar graphs) according to their dual trees, and prove the following formula for the number of triangulations of a convex n-gon whose dual tree has exactly k leaves: n k 2 n−2k n−4 2k−4 C k−2 The proof is bijective and provides a recursive formula for the Catalan numbers similar to, but different from, a classical identity of Touchard. An averaging argument allows one to deduce Touchard's formula from ours.