Asymptotics for the probability of connectedness and the distribution of number of components

Let ρ_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 ρ_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 ρ_n, lim sup ρ_n) in these cases. Only in the convergent case can one have 0 < lim ρ_n < 1. We study the existence of lim ρ_n in this case.