{"paper":{"title":"On random partitions induced by random maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dmitry Krachun, Yuri Yakubovich","submitted_at":"2016-02-03T11:29:07Z","abstract_excerpt":"The lattice of the set partitions of $[n]$ ordered by refinement is studied. Given a map $\\phi: [n] \\rightarrow [n]$, by taking preimages of elements we construct a partition of $[n]$. Suppose $t$ partitions $p_1,p_2,\\dots,p_t$ are chosen independently according to the uniform measure on the set of mappings $[n]\\rightarrow [n]$. The probability that the coarsest refinement of all $p_i$'s is the finest partitions $\\{\\{1\\},\\dots,\\{n\\}\\}$ is shown to approach $1$ for any $t\\geq 3$ and $e^{-1/2}$ for $t=2$. The probability that the finest coarsening of all $p_i$'s is the one-block partition is sho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01270","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}