Episturmian words: a survey

Palindromic Prefixes and Episturmian Words

Let $w$ be an infinite word on an alphabet $A$. We denote by $(n_i)_{i \geq 1}$ the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of $w$. In this text, we give an explicit construction of all words $w$ such that $n_{i+1} \leq 2 n_i + 1$ for any $i$, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity $\limsup n_{i+1}/n_i$, and prove that it is minimal (among all non-periodic words) for the Fibonacci word.

Palindromic Richness

In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of so-called "rich words" is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with X. Droubay and G. Pirillo in 2001. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of w…

A characterization of fine words over a finite alphabet

To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is "fine" if there exists an infinite word u such that, for any lexicographic order, min(w) = au where a = min(A). In this paper, we characterize fine words; specifically, we prove that an infinite word w is fine if and only if w is either a "strict episturmian word" or a strict "skew episturmian word''. This charact…

Introducing Privileged Words: Privileged Complexity of Sturmian Words

In this paper we study the class of so-called privileged words which have been previously considered only a little. We develop the basic properties of privileged words, which turn out to share similar properties with palindromes. Privileged words are studied in relation to previously studied classes of words, rich words, Sturmian words and episturmian words. A new characterization of Sturmian words is given in terms of privileged complexity. The privileged complexity of the Thue-Morse word is also briefly studied.

Extensions of rich words

In [X. Droubay et al, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has at most |w|+1 many distinct palindromic factors, including the empty word. The unified study of words which achieve this limit was initiated in [A. Glen et al, Palindromic richness, Eur. Jour. of Comb. 30 (2009)]. They called these words rich (in palindromes). This article contains several results about rich words and especially extending them. We say that a rich word w can be extended richly with a word u if wu is rich. Some notions are al…

关于回文周期性的若干注记

Some Remarks on Palindromic Periodicities

本文研究有限词的回文周期性，分析多个经典无限词中的回文周期结构，并证明相关性质及最少回文周期性数量的结论。

On Theta-palindromic Richness

In this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism $Θ$. It generalizes the notion of a palindrome into a $Θ$-palindrome -- a word invariant under $Θ$. For languages closed under $Θ$ we give the relation between $Θ$-palindromic complexity and factor complexity. We generalize the notion of richness to $Θ$-richness and we prove analogous characterizations of words that are $Θ$-rich, especially in the case of set of factors invariant under $Θ$. A criterion for $Θ$-richness of $Θ$-episturmian words is given together with other examples of …

On substitutions closed under derivation: examples

We study infinite words fixed by a morphism and their derived words. A derived word is a coding of return words to a factor. We exhibit two examples of sets of morphisms which are closed under derivation --- any derived word with respect to any factor of the fixed point is again fixed by a morphism from this set. The first example involves standard episturmian morphisms, and the second concerns the period doubling morphism.

树状性的代数刻画

Algebraic characterization of dendricity

本文证明了树状移位空间中返回定理的逆命题，从而以返回词构成自由群基这一代数条件完整刻画了树状性。

Rich square-free words

A word w is rich if it has |w|+1 many distinct palindromic factors, including the empty word. A word is square-free if it does not have a factor uu, where u is a non-empty word. Pelantová and Starosta (Discrete Math. 313 (2013)) proved that every infinite rich word contains a square. We will give another proof for that result. Pelantová and Starosta denoted by r(n) the length of a longest rich square-free word on an alphabet of size n. The exact value of r(n) was left as an open question. We will give an upper and a lower bound for r(n), and make a conjecture that our lower bound is exact. We …

Abelian Combinatorics on Words: a Survey

We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these res…

有限字母表上序列的反射复杂度

The reflection complexity of sequences over finite alphabets

本文提出有限字母表上序列的反射复杂度函数，研究其增长性质及与已有复杂度的关系，并用其刻画最终周期序列和 Sturmian 序列，同时分析多类序列及自动序列的反射复杂度。

Short survey about combinatorics on words and algorithmic methods in a ring (draft)

A short survey about combinatorics on words and algorithmic methods in a ring. Special attention is given to Shirshov's results. Adopted for undegraduate students.

Factorizations of the Fibonacci Infinite Word

The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers.

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced.