On typical triangulations of a convex n-gon

Let fn be a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon Pn of n sides. Suppose Tn is a random triangulation, sampled uniformly out of all possible triangulations of n. We study the sum of weights of triangles in Tn and give a general formula for average and variance of this random variable. In addition, we look at several interesting special cases of fn 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 Tn, 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.