Rich, Sturmian, and trapezoidal words

In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced in arXiv:0801.1656 by the second and third authors together with J. Justin and S. Widmer, constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors. Every finite Sturmian word is rich, but not conversely. Trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite Sturmian words. Unfortunately this property does not characterize finite Sturmian words. In this note we show that the only trapezoidal palindromes are Sturmian. More generally we show that Sturmian palindromes can be characterized either in terms of their subword complexity (the trapezoidal property) or in terms of their palindromic complexity. We also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity.Comment: 7 page

Generalized trapezoidal words

The factor complexity function $C_w(n)$ of a finite or infinite word $w$ counts the number of distinct factors of $w$ of length $n$ for each $n \ge 0$. A finite word $w$ of length $|w|$ is said to be trapezoidal if the graph of its factor complexity $C_w(n)$ as a function of $n$ (for $0 \leq n \leq |w|$) is that of a regular trapezoid (or possibly an isosceles triangle); that is, $C_w(n)$ increases by 1 with each $n$ on some interval of length $r$, then $C_w(n)$ is constant on some interval of length $s$, and finally $C_w(n)$ decreases by 1 with each $n$ on an interval of the same length $r$. Necessarily $C_w(1)=2$ (since there is one factor of length $0$, namely the empty word), so any trapezoidal word is on a binary alphabet. Trapezoidal words were first introduced by de Luca (1999) when studying the behaviour of the factor complexity of finite Sturmian words, i.e., factors of infinite "cutting sequences", obtained by coding the sequence of cuts in an integer lattice over the positive quadrant of $\mathbb{R}^2$ made by a line of irrational slope. Every finite Sturmian word is trapezoidal, but not conversely. However, both families of words (trapezoidal and Sturmian) are special classes of so-called "rich words" (also known as "full words") - a wider family of finite and infinite words characterized by containing the maximal number of palindromes - studied in depth by the first author and others in 2009. In this paper, we introduce a natural generalization of trapezoidal words over an arbitrary finite alphabet $\mathcal{A}$, called generalized trapezoidal words (or GT-words for short). In particular, we study combinatorial and structural properties of this new class of words, and we show that, unlike the binary case, not all GT-words are rich in palindromes when $|\mathcal{A}| \geq 3$, but we can describe all those that are rich.Comment: Major revisio

Enumeration and Structure of Trapezoidal Words

Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, \emph{semicentral words}, and show that they are characterized by the property that they can be written as $uxyu$, for a central word $u$ and two different letters $x,y$. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.Comment: Accepted for publication in Theoretical Computer Scienc

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

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 also made about the infinite defect of a word, the number of rich words of length n and two-dimensional rich words.Comment: 19 pages, 3 figure

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

A Classification of Trapezoidal Words

Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open. We show that open trapezoidal words are all primitive and that closed trapezoidal words are all Sturmian. We then show that trapezoidal palindromes are closed (and therefore Sturmian). This allows us to characterize the special factors of Sturmian palindromes. We end with several open problems.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Open and closed complexity of infinite words

In this paper we study the asymptotic behaviour of two relatively new complexity functions defined on infinite words and their relationship to periodicity. Given a factor $w$ of an infinite word $x=x_1x_2x_3\cdots$ with each $x_i$ belonging to a fixed finite set $\mathbb{A},$ we say $w$ is closed if either $w\in \mathbb{A}$ or if $w$ is a complete first return to some factor $v$ of $x.$ Otherwise $w$ is said to be open. We show that for an aperiodic word $x\in \mathbb{A}^\mathbb{N},$ the complexity functions $Cl_x$ (resp. $Op_x)$ that count the number of closed (resp. open) factors of $x$ of each given length are both unbounded. More precisely, we show that if $x$ is aperiodic then $\liminf_{n\in \mathbb{N}} Op_x(n)=+\infty$ and $\limsup_{n\in S} Cl_x(n)=+\infty$ for any syndetic subset $S$ of $\mathbb{N}.$ However, there exist aperiodic infinite words $x$ verifying $\liminf_{n\in \mathbb{N}}Cl_x(n)<+\infty.$ Keywords: word complexity, periodicity, return words

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