Geodesic
Asymptotics for the probability of connectedness and the distribution of number of components
The electronic journal of combinatorics, Tome 7 (2000)
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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 $(\liminf \rho _{n},\;\limsup \rho _{n})$ 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.
DOI : 10.37236/1511
Classification : 05A16, 05C30, 05C40
Mots-clés : probability, connectedness, number of components, asymptotic probabilities, generating function, fraction of structures 
@article{10_37236_1511,
     author = {Jason P. Bell and Edward A. Bender and Peter J. Cameron and L. Bruce Richmond},
     title = {Asymptotics for the probability of connectedness and the distribution of number of components},
     journal = {The electronic journal of combinatorics},
     year = {2000},
     volume = {7},
     doi = {10.37236/1511},
     zbl = {0948.05009},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1511/}
}
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Jason P. Bell; Edward A. Bender; Peter J. Cameron; L. Bruce Richmond. Asymptotics for the probability of connectedness and the distribution of number of components. The electronic journal of combinatorics, Tome 7 (2000). doi: 10.37236/1511

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