## Abstract

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.

Original language | English |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Electronic Journal of Combinatorics |

Volume | 7 |

Issue number | 1 R |

Publication status | Published - 1 Dec 2000 |