On Random Partitions Induced by Random Maps

D. Krachun, Yu Yakubovich

Research output: Contribution to journalArticlepeer-review


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 p1, p2,.. , pt 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 languageEnglish (US)
Pages (from-to)727-740
Number of pages14
JournalJournal of Mathematical Sciences (United States)
Issue number6
StatePublished - Mar 1 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • General Mathematics
  • Applied Mathematics


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