On random partitions induced by random maps

TL;DR

This paper investigates the probabilistic behavior of partitions of a set induced by random maps, analyzing how the coarsest and finest common refinements behave as the number of partitions and the size of the set grow.

Contribution

It provides asymptotic probabilities for the coarsest and finest common refinements of multiple random partitions induced by uniform random maps.

Findings

Probability of coarsest refinement being the finest approaches 1 for t≥3.

Probability of finest coarsening being the one-block partition approaches 1 when t(n)-log n→∞.

Size of the maximal block of the finest coarsening is characterized for fixed t.

Abstract

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 shown to approach 1 if $t(n)-\log{n}\rightarrow \infty$ and $0$ if $t(n)-\log{n}\rightarrow -\infty$. The size of the maximal block of the finest coarsening of all $p_i$'s for a fixed $t$ is also studied.