Set Partition Patterns and the Dimension Index

TL;DR

This paper explores the distribution of the dimension index in avoidance classes of set partitions, revealing new links to noncrossing partitions, 321-avoiding permutations, and classical combinatorial objects.

Contribution

It computes the distribution of the dimension index in specific avoidance classes, establishing novel connections to well-known combinatorial structures.

Findings

Distribution of the dimension index computed for certain avoidance classes

Established links between noncrossing partitions and 321-avoiding permutations

Connected avoidance classes to Motzkin and Fibonacci polynomials

Abstract

The notion of containment and avoidance provides a natural partial ordering on set partitions. Work of Sagan and of Goyt has led to enumerative results in avoidance classes of set partitions, which were refined by Dahlberg et al. through the use of combinatorial statistics. We continue this work by computing the distribution of the dimension index (a statistic arising from the supercharacter theory of finite groups) across certain avoidance classes of partitions. In doing so we obtain a novel connection between noncrossing partitions and 321-avoiding permutations, as well as connections to many other combinatorial objects such as Motzkin and Fibonacci polynomials.