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.

Abstract: In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field, we found that the moments (mean, variance, and higher moments) of novel statistics on set partitions of [n]={1,2,⋯,n} have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in n. This allows exact enumeration of the moments for small n to determine exact formulae for all n.