Asymptotics for the Probability of Connectedness and the Distribution of Number of Components

Let 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 n, provided it exists. It is convenient to distinguish three cases depending on the nature of the power series for the structures: purely formal, convergent on the circle of convergence, and other. We determine all possible values for the pair (lim inf n; lim sup n) in these cases. Only in the convergent case can one have 0 < lim n < 1. We study the existence of lim n in this case. AMS-MOS Subject Classi cation (1990): 05A16; Secondary: 05C30, 05C40