On typical triangulations of a convex $n$-gon

TL;DR

This paper analyzes the statistical properties of weighted triangulations of convex polygons, providing formulas for average and variance, and deriving explicit generating functions for various weight functions, including known and new results.

Contribution

It introduces a systematic approach to study weighted triangulations of convex polygons, generalizing existing results and deriving explicit formulas for diverse weight functions.

Findings

Derived formulas for the average and variance of total triangle weights.

Obtained explicit generating functions for specific weight functions.

Provided new proofs and results on vertex degree, ears, and angle distributions.

Abstract

Let $f_n$ be a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon $P_n$ of $n$ sides. Suppose ${\mathcal T}_n$ is a random triangulation, sampled uniformly out of all possible triangulations of $P_n$. We study the sum of weights of triangles in ${\mathcal T}_n$ and give a general formula for average and variance of this random variable. In addition, we look at several interesting special cases of $f_n$ in which we obtain explicit forms of generating functions for the sum of the weights. For example, among other things, we give new proofs for already known results such as the degree of a fixed vertex and the number of ears in ${\mathcal T}_n,$ as well as, provide new results on the number of "blue" angles and refined information on the distribution of angles at a fixed vertex. We note that our approach is systematic and can be applied to many other new examples while generalizing the existing results.