On the Number of Closed Factors in a Word

A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special. We investigate the structure of closed factors of words. We show that a word of length $n$ contains at least $n+1$ distinct closed factors, and characterize those words having exactly $n+1$ closed factors. Furthermore, we show that a word of length $n$ can contain $\Theta(n^{2})$ many distinct closed factors.Comment: Accepted to LATA 201

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

Palindromes in infinite ternary words

We study infinite words u over an alphabet A satisfying the property P : P(n)+ P(n+1) = 1+ #A for any n in N, where P(n) denotes the number of palindromic factors of length n occurring in the language of u. We study also infinite words satisfying a stronger property PE: every palindrome of u has exactly one palindromic extension in u. For binary words, the properties P and PE coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n)=n+1 for any n in N. In this paper, we focus on ternary infinite words with the language closed under reversal. For such words u, we prove that if C(n)=2n+1 for any n in N, then u satisfies the property P and moreover u is rich in palindromes. Also a sufficient condition for the property PE is given. We construct a word demonstrating that P on a ternary alphabet does not imply PE.Comment: 12 page

Two results on words

The study of combinatorial patterns of words has raised great interest since the early 20th century. In this master's thesis presentation we study two combinatorial patterns. The first pattern is “abelian k-th power free” and the second one is “representability of sets of words of equal length”. For the first pattern we study the context-freeness of non-abelian k-th powers. A word is a non-abelian k-th power if it cannot be factorized in the form w1w2...wk where the wi are permutations of w1 for 2 ≤ i ≤ k. We show that neither the language of non-abelian squares nor the language of non- abelian cubes is context-free. For the second pattern we study the representability of a set of words of fixed length. A set S of words of length n is representable if there exists some word w such that the set of length-n factors of w equals S. We will give lower and upper bounds for the number of such representable sets. Furthermore, we study a variation of the problem: we fix a length t, and try to evaluate the number of sets of words of length n such that there exists some word w of length t such that the set of length-n factors of w equals S. We give a closed-form formula in the case where n ≤ t < 2n. In particular, we give a characterization on two distinct words having the same subset of length-n factors

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

A Note on Symmetries in the Rauzy Graph and Factor Frequencies

We focus on infinite words with languages closed under reversal. If frequencies of all factors are well defined, we show that the number of different frequencies of factors of length n+1 does not exceed 2C(n+1)-2C(n)+1.Comment: 7 page

On Theta-palindromic Richness

In this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism $\Theta$. It generalizes the notion of a palindrome into a $\Theta$-palindrome -- a word invariant under $\Theta$. For languages closed under $\Theta$ we give the relation between $\Theta$-palindromic complexity and factor complexity. We generalize the notion of richness to $\Theta$-richness and we prove analogous characterizations of words that are $\Theta$-rich, especially in the case of set of factors invariant under $\Theta$. A criterion for $\Theta$-richness of $\Theta$-episturmian words is given together with other examples of $\Theta$-rich words.Comment: 14 page

Languages invariant under more symmetries: overlapping factors versus palindromic richness

Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $\mathcal{P}(n) + \mathcal{P}(n+1) \leq 2 + \mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several examples of $G$-rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur