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.

The Permutation Patterns 2007 conference was held 11–15 June 2007 at the University of St Andrews. This was the fifth Permutation Patterns conference; the previous conferences were held at Otago University (Dundein, New Zealand), Malaspina College (Vancouver Island, British Columbia), the University of Florida (Gainesville, Florida), and Reykjavík University (Reykjavík, Iceland). The organizing committee was comprised of Miklós Bóna, Lynn Hynd, Steve Linton, Nik Ruškuc, Einar Steingrímsson, Vincent Vatter, and Julian West. A half-day excursion was taken to Falls of Bruar, Blair Athol Castle and The Queen's View on Loch Tummel.

There were two invited talks:

• Mike Atkinson (Otago University, Dunedin, New Zealand), “Simple permutations and wreath-closed pattern classes”.

• Martin Klazar (Charles University, Prague, Czech Republic), “Polynomial counting”.

There were 35 participants, 23 talks, and a problem session (the problems from which are included at the end of these proceedings). All the main strands of research in permutation patterns were represented, and we hope this is reflected by the articles of these proceedings, especially the eight surveys at the beginning. The conference was supported by the EPSRC and Edinburgh Mathematical Society.

Abstract

We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and provide some new results related to limiting distributions.

Introduction

Monotone subsequences in a permutation p = p1p2 … pn have been the subject of vigorous research for over sixty years. In this paper, we will review three different lines of work. In all of them, we will consider increasing subsequences of a permutation of length n that have a fixed length k. This is in contrast to another line of work, started by Ulam more than sixty years ago, in which the distribution of the longest increasing subsequence of a random permutation has been studied. That direction of research has recently reached a high point in the article of Baik, Deift, and Johansson.

The three directions we consider are distinguished by their definition of monotone subsequences. We can simply require that k entries of a permutation increase from left to right, or we can in addition require that these k entries be in consecutive positions, or we can even require that they be consecutive integers and be in consecutive positions.

Abstract

The enumeration of permutation classes has been accomplished with a variety of techniques. One wide-reaching method is that of enumeration schemes, introduced by Zeilberger and extended by Vatter. In this paper we further extend the method of enumeration schemes to words avoiding permutation patterns. The process of finding enumeration schemes is programmable and allows for the automatic enumeration of many classes of pattern-avoiding words.

Background

The enumeration of permutation classes has been accomplished by many beautiful techniques. One natural extension of permutation classes is pattern-avoiding words. Our concern in this paper is not attractive methods for counting individual classes, but rather developing a systematic technique for enumerating many classes of words. Four main techniques with wide success exist for the systematic enumeration of permutation classes. These are generating trees, insertion encoding, substitution decomposition, and enumeration schemes. In this paper we adapt the method of enumeration schemes, first introduced for permutations by Zeilberger and extended by Vatter to the case of enumerating pattern-restricted words.

Definition 1.1. Let [k]n denote the set of words of length n in the alphabet {1, …, k}, and let w ∈ [k]n, w = w1 … wn. The reduction of w, denoted by red(w), is the unique word of length n obtained by replacing the ith smallest entries of w with i, for each i.