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.

Introduction

An interval of a permutation π corresponds to a set of contiguous indices I = [a, b] such that the set of values π(I) = {π(i) : i ∈ I} is also contiguous. Every permutation of length n has intervals of lengths 0, 1 and n. If a permutation π has no other intervals, then π is said to be simple. For example, the permutation π = 28146357 is not simple as witnessed by the non-trivial interval 4635 (= π(4)π(5)π(6)π(7)), while σ = 51742683 is simple.

While intervals of permutations have applications in biomathematics, particularly to genetic algorithms and the matching of gene sequences (see Corteel, Louchard, and Pemantle for extensive references), simple permutations form the “building blocks” of permutation classes and have thus received intensive study in recent years. We will see in Section 3 the various ways in which simplicity plays a role in the study of permutation classes, but we begin this short survey by introducing the substitution decomposition in Subsection 1.1, and thence by reviewing the structural and enumerative results of simple permutations themselves in Section 2.