Enumeration of Geometric Con gurations on a Convex Polygon

F. Chyzak (ed.), INRIA, (2000), pp. ??{??. Enumeration of Geometric Con gurations on a Convex Polygon Marc Noy Universitat Polit ecnica de Catalunya, Barcelona, Espagne December 16, 1999 Summary by Michel Nguy~̂ en-Tĥ e Abstract We survey recent work on the enumeration of non-crossing con gurations on the set of vertices of a convex polygon, such as triangulations, trees, and forests. Exact formulae and limit laws are determined for several parameters of interest. In the second part of the talk we present results on the enumeration of chord diagrams (pairings of 2n vertices of a convex polygon by means of n disjoint pairs). We present limit laws for the number of components, the size of the largest component and the number of crossings. The use of generating functions and of a variation of Levy's continuity theorem for characteristic functions enable us to establish that most of the limit laws presented here are Gaussian. (Joint work by Marc Noy with Philippe Flajolet and others.) 1. Analytic Combinatorics of Non-crossing Con gurations [3] 1.1. Connected Graphs and General Graphs. Let n = fv1; : : : ; vng be a xed set of points in the plane, conventionally ordered counter-clockwise, that are vertices of a regular n-gon K. De ne a non-crossing graph as a graph with vertex set n whose edges are straight line segments that do not cross. A graph is connected if any two vertices can be joined by a path. Parameters of interest are the number of edges of connected graphs and general graphs, and the number of components of general graphs.