Average values of some Z-parameters in a random set partition

TL;DR

This paper derives exact and asymptotic formulas for various statistics related to set partitions, including Z-parameters, crossings, overlappings, embracings, and pattern occurrences, providing deeper combinatorial insights.

Contribution

It introduces new formulas for average values of multiple set partition statistics, expanding understanding of their asymptotic behavior and distribution.

Findings

Exact formulas for average Z-parameters and related statistics.

Asymptotic formulas for large set partitions.

Insights into the distribution of crossings, overlappings, and pattern occurrences.

Abstract

We find exact and asymptotic formulas for the average values of several statistics on set partitions: of Carlitz's $q$-Stirling distributions, of the numbers of crossings in linear and circular representations of set partitions, of the numbers of overlappings and embracings, and of the numbers of occurrences of a 2-pattern.