A Survey of Simple Permutations

TL;DR

This survey reviews recent enumerative and structural results on simple permutations, highlighting their significance in permutation class theory and their special properties when finitely many are involved.

Contribution

It compiles recent findings on simple permutations and elucidates their role in permutation class properties and enumeration.

Findings

Classes with finitely many simple permutations have unique structural properties.

Simple permutations are central to understanding permutation class enumeration.

Finitely based classes with simple permutations exhibit special ordering properties.

Abstract

We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study of permutation classes. We demonstrate how classes containing only finitely many simple permutations satisfy a number of special properties relating to enumeration, partial well-order and the property of being finitely based.