Harmonic and gold Sturmian words

AbstractIn the combinatorics of Sturmian words an essential role is played by the set PER of all finite words w on the alphabet A={a,b} having two periods p and q which are coprime and such that |w|=p+q−2. As is well known, the set St of all finite factors of all Sturmian words equals the set of factors of PER. Moreover, the elements of PER have many remarkable structural properties. In particular, the relation Stand=A∪PER{ab,ba} holds, where Stand is the set of all finite standard Sturmian words. In this paper we introduce two proper subclasses of PER that we denote by Harm and Gold. We call an element of Harm a harmonic word and an element of Gold a gold word. A harmonic word w beginning with the letter x is such that the ratio of two periods p/q, with p<q, is equal to its slope, i.e., (|w|y+1)/(|w|x+1), where {x,y}={a,b}. A gold word is an element of PER such that p and q are primes. Some characterizations of harmonic words are given and the number of harmonic words of each length is computed. Moreover, we prove that St is equal to the set of factors of Harm and to the set of factors of Gold. We introduce also the classes Harm and Gold of all infinite standard Sturmian words having infinitely many prefixes in Harm and Gold, respectively. We prove that Gold∩Harm contain continuously many elements. Finally, some conjectures are formulated

A Coloring Problem for Sturmian and Episturmian Words

We consider the following open question in the spirit of Ramsey theory: Given an aperiodic infinite word $w$, does there exist a finite coloring of its factors such that no factorization of $w$ is monochromatic? We show that such a coloring always exists whenever $w$ is a Sturmian word or a standard episturmian word

Characterizations of finite and infinite episturmian words via lexicographic orderings

In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a "wide sense" (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2-letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian.Comment: 18 pages; to appear in the European Journal of Combinatoric

Open and Closed Prefixes of Sturmian Words

A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We deal with the sequence of open and closed prefixes of Sturmian words and prove that this sequence characterizes every finite or infinite Sturmian word up to isomorphisms of the alphabet. We then characterize the combinatorial structure of the sequence of open and closed prefixes of standard Sturmian words. We prove that every standard Sturmian word, after swapping its first letter, can be written as an infinite product of squares of reversed standard words.Comment: To appear in WORDS 2013 proceeding

The sequence of open and closed prefixes of a Sturmian word

A finite word is closed if it contains a factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We are interested in the {\it oc-sequence} of a word, which is the binary sequence whose $n$-th element is $0$ if the prefix of length $n$ of the word is open, or $1$ if it is closed. We exhibit results showing that this sequence is deeply related to the combinatorial and periodic structure of a word. In the case of Sturmian words, we show that these are uniquely determined (up to renaming letters) by their oc-sequence. Moreover, we prove that the class of finite Sturmian words is a maximal element with this property in the class of binary factorial languages. We then discuss several aspects of Sturmian words that can be expressed through this sequence. Finally, we provide a linear-time algorithm that computes the oc-sequence of a finite word, and a linear-time algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of arXiv:1306.225

Extremal properties of (epi)Sturmian sequences and distribution modulo 1

Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in distribution of real numbers modulo 1 via combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some of which have been rediscovered several times

On the Structure of Bispecial Sturmian Words

A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic} bispecial Sturmian words are precisely the maximal internal factors of \emph{primitive} Christoffel words. We extend this result by showing that bispecial Sturmian words are precisely the maximal internal factors of \emph{all} Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167

On prefixal factorizations of words

We consider the class ${\cal P}_1$ of all infinite words $x\in A^\omega$ over a finite alphabet $A$ admitting a prefixal factorization, i.e., a factorization $x= U_0 U_1U_2 \cdots$ where each $U_i$ is a non-empty prefix of $x.$ With each $x\in {\cal P}_1$ one naturally associates a "derived" infinite word $\delta(x)$ which may or may not admit a prefixal factorization. We are interested in the class ${\cal P}_{\infty}$ of all words $x$ of ${\cal P}_1$ such that $\delta^n(x) \in {\cal P}_1$ for all $n\geq 1$. Our primary motivation for studying the class ${\cal P}_{\infty}$ stems from its connection to a coloring problem on infinite words independently posed by T. Brown in \cite{BTC} and by the second author in \cite{LQZ}. More precisely, let ${\bf P}$ be the class of all words $x\in A^\omega$ such that for every finite coloring $\varphi : A^+ \rightarrow C$ there exist $c\in C$ and a factorization $x= V_0V_1V_2\cdots$ with $\varphi(V_i)=c$ for each $i\geq 0.$ In \cite{DPZ} we conjectured that a word $x\in {\bf P}$ if and only if $x$ is purely periodic. In this paper we show that ${\bf P}\subseteq {\cal P}_{\infty},$ so in other words, potential candidates to a counter-example to our conjecture are amongst the non-periodic elements of ${\cal P}_{\infty}.$ We establish several results on the class ${\cal P}_{\infty}$. In particular, we show that a Sturmian word $x$ belongs to ${\cal P}_{\infty}$ if and only if $x$ is nonsingular, i.e., no proper suffix of $x$ is a standard Sturmian word