## Abstract

The partition lattice of the set [n] with respect to refinement is studied. Any map ϕ: [n] → [n], is associated with a partition of [n] by taking preimages of the elements. Assume that t partitions p_{1}, p_{2},.. , p_{t} are chosen independently according to the uniform measure on the set of mappings [n] → [n]. It is shown that the probability for the coarsest refinement of all the partitions pi to be the finest partition {{1},.. , {n}} tends to 1 for any t ≥ 3 and to e^{−1/2} for t = 2. It is also proved that the probability for the finest coarsening of the partitions pi to be the one-block partition tends to 1 as t(n) − log n→∞ and tends to 0 as t(n) − log n. The size of the maximal block of the finest coarsening of all the pi for a fixed t is also studied.

Original language | English (US) |
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Pages (from-to) | 727-740 |

Number of pages | 14 |

Journal | Journal of Mathematical Sciences (United States) |

Volume | 229 |

Issue number | 6 |

DOIs | |

State | Published - Mar 1 2018 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- General Mathematics
- Applied Mathematics