Triangulating the Circle, at Random

(2m - 2)! cm - (m-1)!m! One of interesting aspects of Polya's paper is that it exposed readers to his newly developed theory of figurate series. We wish to consider idea of letting n -*> oo and studying triangulations of oo-gon, i.e. circle. This question doesn't make much sense as combinatorics, but we can shift viewpoint and consider random triangulations of n-gon, in which each of cn-1 possible triangulations is equally likely. The purpose of this paper is to show that there exists an object the random triangulation of a circle which is in a natural sense n -* oo limit of random triangulation of n-gon. As with Polya [9], exposition takes readers into some newly developed theory of author. Let's start by recalling a precise definition. A triangulation of a finite set S is a collection of nonintersecting line segments with endpoints from S such that convex hull of S is partitioned into triangular regions. We shall be concerned only with cases Sn consisting of vertices of regular n-gon inscribed in a fixed