Combinatorics on words is a field that has grown separately within several branches of mathematics, such as number theory, group theory or probability theory, and appears frequently in problems of theoretical computer science, as dealing with automata and formal languages.

A unified treatment of the theory appeared in Lothaire's Combinatorics on Words. Since then, the field has grown rapidly. This book presents new topics of combinatorics on words.

Several of them were not yet ripe for exposition, or even not yet explored, twenty years ago. The spirit of the book is the same, namely an introductory exposition of a field, with full proofs and numerous examples, and further developments deferred to problems, or mentioned in the Notes.

This book is independent of Lothaire's first book, in the sense that no knowledge of the first volume is assumed. In order to avoid repetitions, some results of the first book, when needed here, are explicitly quoted, and are only referred for the proof to the first volume.

This volume presents, compared with the previous one, two important new features. It is first of all a complement in the sense that it goes deeper in the same direction. For example, the theory of unavoidable patterns (Chapter 3) is a generalization of the theory of square-free words and morphisms. In the same way, the chapters on statistics on words and permutations (Chapters 10 and 11) are a continuation of the chapter on transformations on words of the previous volume.

Introduction

The aim of this chapter is to provide an introduction to several concepts used elsewhere in the book. It fixes the general notation on words used elsewhere. It also introduces more specialized notions of general interest. For instance, the notion of a uniformly recurrent word used in several other chapters is introduced here.

We start with the notation concerning finite and infinite words. We also describe the Cantor space topology on the space of infinite words.

We provide a basic introduction to the theory of automata. It covers the determinization algorithm, part of Kleene's theorem, syntactic monoids and basic facts about transducers. These concepts are illustrated on the classical combinatorial examples of the de Bruijn graph, and the Morse-Hedlund theorem.

We also consider the relationship with generating series, as a useful tool for the enumeration of words.

We introduce some basic concepts of symbolic dynamical systems, in relation with automata. We prove the equivalence between the notions of minimality and uniform recurrence. Entropy is considered, and we show how to compute it for a sofic system.

We also present a more specialized subject, namely unavoidable sets. This notion is easy to define but leads to interesting and significant results. In this sense, the last section of this chapter is a foretaste of the rest of the book.

Introduction

In this chapter we briefly summarize some of the important results in information theory which we have not been able to treat in detail. We shall give no proofs, but instead refer the interested reader elsewhere, usually to a textbook, sometimes to an original paper, for details.

We choose to restrict our attention solely to generalizations and extensions of the twin pearls of information theory, Shannon's channel coding theorem (Theorem 2.4 and its corollary) and his source coding theorem (Theorem 3.4). We treat each in a separate section.

The channel coding theorem

We restate the theorem for reference (see Corollary to Theorem 2.4).

Associated with each discrete memoryless channel, there is a nonnegative number C (called channel capacity) with the following property. For any ε > 0 and R < C, for large enough n, there exists a code of length n and rate ≧ R (i.e., with at least 2Rn distinct codewords), and an appropriate decoding algorithm, such that, when the code is used on the given channel, the probability of decoder error is < ε.

We shall now conduct a guided tour through the theorem, pointing out as we go places where the hypotheses can be weakened or the conclusions strengthened. The points of interest will be the phrases discrete memoryless channel, a nonnegative number C, for large enough n and there exists a code … and … decoding algorithm. We shall also briefly discuss various converses to the coding theorem.

Introduction

This chapter is devoted, as was the previous one, to combinatorial properties of permutations, considered as words. The starting point in this subject is the bivalent status of permutations, which can be considered as products of cycles as well as a sequence of the first n integers written in disorder.

The fundamental results concerning this area are presented in Chapter 10 of Lothaire 1983. They consist essentially in two transformations on words. The first one (first fundamental transformation) is an encoding of the cycle decomposition of a permutation. The second one (second fundamental transformation) accounts for a statistical property of permutations, namely the equidistribution of the number of inversions and the inverse major index on permutations with a given shape.

In the previous chapter (Chapter 10) some other properties are presented, including basic facts on q-calculus and additional statistics on permutations.

In this chapter, we carry on with complements focused on two main aspects. The first one is a shortcut avoiding the second fundamental transformation by a simple evaluation of determinants. The second one is the analogy between the first fundamental transformation and the Lyndon factorization (the Gessel normalization).

The organization of the chapter is the following. After some preliminaries, Section 11.2 provides a determinantal expression for the commutative image of some sets of words. This expression is used in Sections 11.3 and 11.4, for evaluating respectively the inverse major index and the number of inversions of permutations with a given shape. In Section 11.5 the Gessel normalization is introduced. In Section 11.6, it is then applied to evaluating the major index of permutations with a given cycle structure.

Introduction

Sturmian words are infinite words over a binary alphabet that have exactly n + 1 factors of length n for each n ≥ 0. It appears that these words admit several equivalent definitions, and can even be described explicitly in arithmetic form. This arithmetic description is a bridge between combinatorics and number theory. Moreover, the definition by factors makes Sturmian words define symbolic dynamical systems. The first detailed investigations of these words were done from this point of view. Their numerous properties and equivalent definitions, and also the fact that the Fibonacci word is Sturmian, have led to a great development, under various terminologies, of the research.

The aim of this chapter is to present basic properties of Sturmian words and of their transformation by morphisms. The style of exposition relies basically on combinatorial arguments.

The first section is devoted to the proof of the Morse–Hedlund theorem stating the equivalence of Sturmian words with the set of balanced aperiodic words and the set of mechanical words of irrational slope. We also mention several other formulations of mechanical words, such as rotations and cutting sequences. We next give properties of the set of factors of one Sturmian word, such as closure under reversal, the minimality of the associated dynamical system, the fact that the set depends only on the slope, and we give the description of special words.