# Non-crossing partitions

## TL;DR

This paper explores the concept of non-crossing partitions, their historical significance in combinatorics, and their recent applications across various mathematical fields, including group theory and topology.

## Contribution

It reviews the development of non-crossing partitions and their generalizations related to Coxeter and Artin groups, highlighting their diverse mathematical connections.

## Key findings

- Connections to free probability and braid groups
- Introduction of analogues of the non-crossing partition lattice
- Association with Coxeter and Artin groups of type A

## Abstract

Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical non-crossing partitions are associated to Coxeter and Artin groups of type $\mathsf{A}_n$, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.