Central Limit Theorems for some Set Partition Statistics

TL;DR

This paper establishes the normal distribution behavior for certain statistics of set partitions, including crossings, dimension index, and levels, using a novel stochastic approach.

Contribution

It introduces a new stochastic representation to prove the conjectured normality of set partition statistics and extends CLTs to multiple related measures.

Findings

Number of crossings in a random set partition is normally distributed.

Central limit theorems are proved for the dimension index and number of levels.

A novel stochastic representation is developed for these proofs.

Abstract

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index and the number of levels.