Asymptotics for the probability of connectedness and the distribution of number of components
Summary: Let \(\rho_n\) be the fraction of structures of ``size'' \(n\) which are ``connected''; e.g., (a) the fraction of labeled or unlabeled \(n\)-vertex graphs having one component, (b) the fraction of partitions of \(n\) or of an \(n\)-set having a single part or block, or (c) the fraction of \(n\)-vertex forests that contain only one tree. Various authors have considered \(\lim \rho_n\), provided it exists. It is convenient to distinguish three cases depending on the nature of the power series for the structures: pure formal, convergent on the circle of convergence, and other. We determine all possible values for the pair (\(\lim\inf\rho_n\), \(\limsup\rho_n\)) in these cases. Only in the convergent case can one have \(0<\lim \rho_n<1\). We study the existence of \(\lim\rho_n\) in this case.
scientific article; zbMATH DE number 1467491