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{\bf Jason P. Bell, Edward A. Bender, Peter J. Cameron,\hfil\break
L. Bruce Richmond }
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{\bf Asymptotics for the Probability of Connectedness and the Distribution of Number of Components}
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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: purely formal, convergent on
the circle of convergence, and other.
We determine all possible values for the pair
\hbox{$(\liminf \rho _{n},\;\limsup \rho _{n})$} 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.
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