The Probability of Connectedness of Random Structures

Let $a(n)$ be the number of $n$-vertex labeled or unlabeled structures and let $c(n)$ be the number which are connected. Wright, Compton, and others have examined the limiting behavior of $c(n)/a(n)$, primarily when the limit exists and is 0 or 1. We revisit these cases and also look at the case when the limit is strictly between 0 and 1. The results are closely tied to the radius of convergence $r$ of the associated power series, whether the series converges at $r$, and the behavior of the number of 2-component structures. We also consider the liminf and limsup of $c(n)/a(n)$. This is work in progress with Bruce Richmond and Peter Cameron.