scholarly journals Words with many Palindrome Pair Factors

10.37236/5583 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Adam Borchert ◽  
Narad Rampersad
Keyword(s):  
Infinite Words ◽  
Sturmian Word ◽  
Proper Factor ◽  
Sturmian Words

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many $n$, every length-$n$ factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue—Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.

scholarly journals The Maximal Number of Runs in Standard Sturmian Words

10.37236/2473 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Paweł Baturo ◽  
Marcin Piątkowski ◽  
Wojciech Rytter
Keyword(s):  
Explicit Formula ◽  
Special Class ◽  
Linear Time ◽  
Infinite Sequence ◽  
Integer Sequence ◽  
Sturmian Word ◽  
Compact Representations ◽  
Sturmian Words ◽  
Standard Word

We investigate some repetition problems for a very special class $\mathcal{S}$ of strings called the standard Sturmian words, which  have very compact representations in terms of sequences of integers. Usually the size of this word is exponential with respect to the size of its integer sequence, hence we are dealing with repetition problems in compressed strings. An explicit formula is given for the number $\rho(w)$ of runs in a standard word $w$. We show that $\rho(w)/|w|\le 4/5$ for each $w\in S$, and  there is an infinite sequence of strictly growing words $w_k\in {\mathcal{S}}$ such that $\lim_{k\rightarrow \infty} \frac{\rho(w_k)}{|w_k|} = \frac{4}{5}$. Moreover, we show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation.

scholarly journals An Inequality for the Number of Periods in a Word

2021 ◽  
pp. 1-18
Author(s):  
Daniel Gabric ◽  
Narad Rampersad ◽  
Jeffrey Shallit
Keyword(s):  
Critical Exponent ◽  
Infinite Words ◽  
Sturmian Word ◽  
Special Cases

We prove an inequality for the number of periods in a word [Formula: see text] in terms of the length of [Formula: see text] and its initial critical exponent. Next, we characterize all periods of the length-[Formula: see text] prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of [Formula: see text], and use this result to show that our inequality is tight for infinitely many words [Formula: see text]. We propose two related measures of periodicity for infinite words. Finally, we also consider special cases where [Formula: see text] is overlap-free or squarefree.

scholarly journals RECENT RESULTS ON EXTENSIONS OF STURMIAN WORDS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 371-385 ◽  
Author(s):  
JEAN BERSTEL
Keyword(s):  
Infinite Words ◽  
Combinatorial Properties ◽  
The Past ◽  
Letter Alphabet ◽  
Sturmian Words

Sturmian words are infinite words over a two-letter alphabet that admit a great number of equivalent definitions. Most of them have been given in the past ten years. Among several extensions of Sturmian words to larger alphabets, the Arnoux–Rauzy words appear to share many of the properties of Sturmian words. In this survey, combinatorial properties of these two families are considered and compared.

scholarly journals Rigidity and Substitutive Dendric Words

2018 ◽  
Vol 29 (05) ◽  
pp. 705-720 ◽  
Author(s):  
V. Berthé ◽  
F. Dolce ◽  
F. Durand ◽  
J. Leroy ◽  
D. Perrin
Keyword(s):  
Bipartite Graphs ◽  
Constant Length ◽  
Infinite Words ◽  
Algebraic Properties ◽  
Interval Exchanges ◽  
Left And Right ◽  
Sturmian Words

Dendric words are infinite words that are defined in terms of extension graphs. These are bipartite graphs that describe the left and right extensions of factors. Dendric words are such that all their extension graphs are trees. They are also called tree words. This class of words includes classical families of words such as Sturmian words, codings of interval exchanges, or else, Arnoux–Rauzy words. We investigate here the properties of substitutive dendric words and prove some rigidity properties, that is, algebraic properties on the set of substitutions that fix a dendric word. We also prove that aperiodic minimal dendric subshifts (generated by dendric words) cannot have rational topological eigenvalues, and thus, cannot be generated by constant length substitutions.

ON THE PALINDROMIC COMPLEXITY OF INFINITE WORDS

2004 ◽  
Vol 15 (02) ◽  
pp. 293-306 ◽  
Author(s):  
S. BRLEK ◽  
S. HAMEL ◽  
M. NIVAT ◽  
C. REUTENAUER
Keyword(s):  
Mild Condition ◽  
Infinite Words ◽  
Finite Set ◽  
Sturmian Words ◽  
Check Condition

We study the problem of constructing infinite words having a prescribed finite set P of palindromes. We first establish that the language of all words with palindromic factors in P is rational. As a consequence we derive that there exists, with some additional mild condition, infinite words having P as palindromic factors. We prove that there exist periodic words having the maximum number of palindromes as in the case of Sturmian words, by providing a simple and easy to check condition. Asymmetric words, those that are not the product of two palindromes, play a fundamental role and an enumeration is provided.

scholarly journals AVOIDING ABELIAN POWERS IN BINARY WORDS WITH BOUNDED ABELIAN COMPLEXITY

2011 ◽  
Vol 22 (04) ◽  
pp. 905-920 ◽  
Author(s):  
JULIEN CASSAIGNE ◽  
GWÉNAËL RICHOMME ◽  
KALLE SAARI ◽  
LUCA Q. ZAMBONI
Keyword(s):  
Infinite Number ◽  
Open Problem ◽  
Orbit Closure ◽  
Infinite Words ◽  
Morphic Image ◽  
In The Beginning ◽  
Sturmian Words ◽  
Free Word ◽  
Recurrent Word

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian k-powers for some integer k, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian k-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian k-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo–Varricchio and Halbeisen–Hungerbühler.

scholarly journals On the number of factors in codings of three interval exchange

2011 ◽  
Vol Vol. 13 no. 3 (Graph Theory) ◽  
Author(s):  
Petr Ambrož ◽  
Anna Frid ◽  
Zuzana Masáková ◽  
Edita Pelantová
Keyword(s):  
Graph Theory ◽  
Asymptotic Formula ◽  
Interval Exchange ◽  
Infinite Words ◽  
Number Of Factors ◽  
Strong Relation ◽  
International Audience ◽  
Sturmian Words

Graph Theory International audience We consider exchange of three intervals with permutation (3, 2, 1). The aim of this paper is to count the cardinality of the set 3iet (N) of all words of length N which appear as factors in infinite words coding such transformations. We use the strong relation of 3iet words and words coding exchange of two intervals, i.e., Sturmian words. The known asymptotic formula #2iet(N)/N-3 similar to 1/pi(2) for the number of Sturmian factors allows us to find bounds 1/3 pi(2) +o(1) \textless= #3iet(N)N-4 \textless= 2 pi(2) + o(1)

scholarly journals Sumsets of Finite Beatty Sequences

10.37236/1614 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
Jane Pitman
Keyword(s):  
Arithmetic Progression ◽  
Continued Fractions ◽  
Sturmian Word ◽  
Beatty Sequences ◽  
Sturmian Words ◽  
Beatty Sequence

An investigation of the size of $S+S$ for a finite Beatty sequence $S=(s_i)=(\lfloor i\alpha+\gamma \rfloor)$, where $\lfloor \hphantom{x} \rfloor$ denotes "floor", $\alpha$, $\gamma$ are real with $\alpha\ge 1$, and $0\le i \le k-1$ and $k\ge 3$. For $\alpha>2$, it is shown that $|S+S|$ depends on the number of "centres" of the Sturmian word $\Delta S=(s_i-s_{i-1})$, and hence that $3(k-1)\le |S+S|\le 4k-6$ if $S$ is not an arithmetic progression. A formula is obtained for the number of centres of certain finite periodic Sturmian words, and this leads to further information about $|S+S|$ in terms of finite nearest integer continued fractions.

scholarly journals A Square Root Map on Sturmian Words

10.37236/6074 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Jarkko Peltomäki ◽  
Markus A. Whiteland
Keyword(s):  
Fixed Points ◽  
General Setting ◽  
Square Root ◽  
Sturmian Word ◽  
Word Equation ◽  
Sturmian Words ◽  
Infinite Set ◽  
Curious Property

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope $\alpha$, there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper prefix). A Sturmian word $s$ of slope $\alpha$ can be written as a product of these six minimal squares: $s = X_1^2 X_2^2 X_3^2 \cdots$. The square root of $s$ is defined to be the word $\sqrt{s} = X_1 X_2 X_3 \cdots$. The main result of this paper is that $\sqrt{s}$ is also a Sturmian word of slope $\alpha$. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of $\sqrt{s}$ and an occurrence of any prefix of $\sqrt{s}$ in $s$. Related to the square root map, we characterize the solutions of the word equation $X_1^2 X_2^2 \cdots X_n^2 = (X_1 X_2 \cdots X_n)^2$ in the language of Sturmian words of slope $\alpha$ where the words $X_i^2$ are minimal squares of slope $\alpha$.We also study the square root map in a more general setting. We explicitly construct an infinite set of non-Sturmian fixed points of the square root map. We show that the subshifts $\Omega$ generated by these words have a curious property: for all $w \in \Omega$ either $\sqrt{w} \in \Omega$ or $\sqrt{w}$ is periodic. In particular, the square root map can map an aperiodic word to a periodic word.

scholarly journals Quasi-Sturmian colorings on regular trees

2019 ◽  
Vol 40 (12) ◽  
pp. 3403-3419
Author(s):  
DONG HAN KIM ◽  
SEUL BEE LEE ◽  
SEONHEE LIM ◽  
DEOKWON SIM
Keyword(s):  
Infinite Words ◽  
Different Types ◽  
Sturmian Words

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

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