Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:Enumerative problems inspired by Mayer’s theory of cluster integrals. | Enumerative problems inspired by Mayer s theory of cluster integrals Pierre Leroux Departement de Mathematiques et LaCIM Universite du Quebec a Montreal Canada leroux@ Submitted Jul 31 2003 Accepted Apr 20 2004 Published May 14 2004 MR Subject Classifications 05A15 05C05 05C30 82Axx Abstract The basic functional equations for connected and 2-connnected graphs can be traced back to the statistical physicists Mayer and Husimi. They play an essential role in establishing rigorously the virial expansion for imperfect gases. We first review these functional equations putting the emphasis on the structural relationships between the various classes of graphs. We then investigate the problem of enumerating some classes of connected graphs all of whose 2-connected components blocks are contained in a given class B. Included are the species of Husimi graphs B complete graphs cacti B unoriented cycles and oriented cacti B oriented cycles . For each of these we address the question of their labelled and unlabelled enumeration according or not to their block-size distributions. Finally we discuss the molecular expansion of these species. It consists of a descriptive classification of the unlabelled structures in terms of elementary species from which all their symmetries can be deduced. 1 Introduction Functional equations for connected graphs and blocks Informally a combinatorial species is a class of labelled discrete structures which is closed under isomorphisms induced by relabelling along bijections. See Joyal 13 and Bergeron Labelle Leroux 2 for an introduction to the theory of species. Note that the present article is mostly self-contained. To each species F are associated a number of series With the partial support of FQRNT Quebec and CRSNG Canada THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R32 1 expansions among which are the following. The exponential generating function F x for labelled enumeration is defined by F x X F n x n 1 where F n denotes .
