Closed expressions for averages of set partition statistics

TL;DR

This paper derives closed-form expressions for the moments of various set partition statistics, revealing their structure as linear combinations of shifted Bell numbers with polynomial coefficients, enabling exact enumeration for small n.

Contribution

It introduces new closed-form formulas for moments of set partition statistics, expanding understanding of their enumerative properties and providing exact formulas for all n.

Findings

Moments are linear combinations of shifted Bell numbers.

Coefficients are polynomials in n.

Exact enumeration possible for small n.

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 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 $n$.