What Could Be a Simple Permutation?

TL;DR

This paper explores various definitions of simple permutations, introduces new types, and analyzes their interrelations along with asymptotic and geometric properties of these classes.

Contribution

It defines new types of simple permutations and studies their connections and properties, expanding understanding of permutation structures.

Findings

Multiple definitions of simple permutations are compared.

New types of simple permutations are introduced.

Asymptotic and geometrical properties of these classes are analyzed.

Abstract

Different ways to describe a permutation, as a sequence of integers, or a product of Coxeter generators, or a tree, give different choices to define a simple permutation. We recollect few of them, define new types of simple permutations, and analyze their interconnections and some asymptotic and geometrical properties of these classes.