Analytic Combinatorics of Non-crossing Conngurations Analytic Combinatorics of Non-crossing Conngurations Analytic Combinatorics of Non-crossing Conngurations

This paper describes a systematic approach to the enumeration of \non-crossing" geometric conngurations built on vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc. Combinatoire analytique des conngurations sans croisement R esum e : Cet article d ecrit une approche syst ematique au d enombrement de conngu-rations g eom etriques \sans croisements" construites sur les sommets d'un n-gone convexe plan. L'approche repose sur les fonctions g en eratrices, les m ethodes symboliques, l'analyse de singularit es et la perturbation de singularit es. On en d eduit des r esultats tant exacts qu'asymptotiques pour arbres, for^ ets, graphes connexes et g en eraux, dissections et partitions. Des lois limites de formes gaussienne r esultent egalement de cette m ethode; elles concernent le nombre de feuilles dans les arbres, le nombre de composantes ou d'ar^ etes dans les graphes, etc. Abstract This paper describes a systematic approach to the enumeration of \non-crossing" geometric conngurations built on vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc.