scholarly journals Doubled Patterns are 3-Avoidable

10.37236/5618 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Pascal Ochem
Keyword(s):  
Combinatorics On Words ◽  
Letter Alphabet ◽  
Infinite Word

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f=h(p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. A pattern is said to be doubled if no variable occurs only once. Doubled patterns with at most 3 variables and doubled patterns with at least 6 variables are $3$-avoidable. We show that doubled patterns with 4 and 5 variables are also $3$-avoidable.

scholarly journals Application of Entropy Compression in Pattern Avoidance

10.37236/3038 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Pascal Ochem ◽  
Alexandre Pinlou
Keyword(s):  
Positive Answer ◽  
Pattern Avoidance ◽  
Algebraic Combinatorics ◽  
Combinatorics On Words ◽  
Letter Alphabet ◽  
Infinite Word ◽  
Entropy Compression

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f= h(p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We give a positive answer to Problem 3.3.2 in Lothaire's book "Algebraic combinatorics on words'", that is, every pattern with $k$ variables of length at least $2^k$ (resp. $3\times2^{k-1}$) is 3-avoidable (resp. 2-avoidable). This conjecture was first stated by Cassaigne in his thesis in 1994. This improves previous bounds due to Bell and Goh, and Rampersad.

scholarly journals Avoidability of Formulas with Two Variables

10.37236/6536 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Pascal Ochem ◽  
Matthieu Rosenfeld
Keyword(s):  
Combinatorics On Words ◽  
Letter Alphabet ◽  
Infinite Word

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We consider the patterns such that at most two variables appear at least twice,  or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is $2$-avoidable, and if it is $2$-avoidable, we determine whether it is avoided by exponentially many binary words.

scholarly journals Combinatorial aspects of Escher tilings

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Alexandre Blondin Massé ◽  
Srecko Brlek ◽  
Sébastien Labbé
Keyword(s):  
Mathematical Community ◽  
Combinatorics On Words ◽  
Combinatorial Object ◽  
Square Grid ◽  
Letter Alphabet ◽  
International Audience ◽  
Escher Tilings

International audience In the late 30's, Maurits Cornelis Escher astonished the artistic world by producing some puzzling drawings. In particular, the tesselations of the plane obtained by using a single tile appear to be a major concern in his work, drawing attention from the mathematical community. Since a tile in the continuous world can be approximated by a path on a sufficiently small square grid - a widely used method in applications using computer displays - the natural combinatorial object that models the tiles is the polyomino. As polyominoes are encoded by paths on a four letter alphabet coding their contours, the use of combinatorics on words for the study of tiling properties becomes relevant. In this paper we present several results, ranging from recognition of these tiles to their generation, leading also to some surprising links with the well-known sequences of Fibonacci and Pell. Lorsque Maurits Cornelis Escher commença à la fin des années 30 à produire des pavages du plan avec des tuiles, il étonna le monde artistique par la singularité de ses dessins. En particulier, les pavages du plan obtenus avec des copies d'une seule tuile apparaissent souvent dans son œuvre et ont attiré peu à peu l'attention de la communauté mathématique. Puisqu'une tuile dans le monde continu peut être approximée par un chemin sur un réseau carré suffisamment fin - une méthode universellement utilisée dans les applications utilisant des écrans graphiques - l'objet combinatoire qui modèle adéquatement la tuile est le polyomino. Comme ceux-ci sont naturellement codés par des chemins sur un alphabet de quatre lettres, l'utilisation de la combinatoire des mots devient pertinente pour l'étude des propriétés des tuiles pavantes. Nous présentons dans ce papier plusieurs résultats, allant de la reconnaissance de ces tuiles à leur génération, conduisant à des liens surprenants avec les célèbres suites de Fibonacci et de Pell.

scholarly journals Critical Exponents of Words over 3 Letters

10.37236/612 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Elise Vaslet
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Letter Alphabet ◽  
Infinite Word

For all $\alpha \geq RT(3)$ (where $RT(3) = 7/4$ is the repetition threshold for the $3$-letter alphabet), there exists an infinite word over 3 letters whose critical exponent is $\alpha$.

scholarly journals On Repetition Thresholds of Caterpillars and Trees of Bounded Degree

10.37236/6793 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Borut Lužar ◽  
Pascal Ochem ◽  
Alexandre Pinlou
Keyword(s):  
Real Number ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Letter Alphabet ◽  
Graph Classes ◽  
Infinite Word

The repetition threshold is the smallest real number $\alpha$ such that there exists an infinite word over a $k$-letter alphabet that avoids repetition of exponent strictly greater than $\alpha$. This notion can be generalized to graph classes. In this paper, we completely determine the repetition thresholds for caterpillars and caterpillars of maximum degree $3$. Additionally, we present bounds for the repetition thresholds of trees with bounded maximum degrees.

scholarly journals Avoiding conjugacy classes on the 5-letter alphabet

2020 ◽  
Vol 54 ◽  
pp. 2
Author(s):  
Golnaz Badkobeh ◽  
Pascal Ochem
Keyword(s):  
Conjugacy Class ◽  
Cyclic Permutation ◽  
Conjugacy Classes ◽  
Letter Alphabet ◽  
Infinite Word

We construct an infinite word w over the 5-letter alphabet such that for every factor f of w of length at least two, there exists a cyclic permutation of f that is not a factor of w. In other words, w does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [Theoret. Comput. Sci. 726 (2018) 1–4].

scholarly journals Permutations realized by shifts

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Sergi Elizalde
Keyword(s):  
Relative Order ◽  
Letter Alphabet ◽  
Infinite Word ◽  
Set Of Permutations ◽  
International Audience ◽  
Forbidden Patterns ◽  
Pattern Containment

International audience A permutation $\pi$ is realized by the shift on $N$ symbols if there is an infinite word on an $N$-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as $\pi$. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [J.M. Amigó, S. Elizalde and M. Kennel, $\textit{J. Combin. Theory Ser. A}$ 115 (2008), 485―504] that the shortest forbidden patterns of the shift on $N$ symbols have length $N+2$. In this paper we give a characterization of the set of permutations that are realized by the shift on $N$ symbols, and we enumerate them with respect to their length. Une permutation $\pi$ est réalisée par le $\textit{shift}$ avec $N$ symboles s'il y a un mot infini sur un alphabet de $N$ lettres dont les déplacements successifs d'une position à gauche sont lexicographiquement dans le même ordre relatif que $\pi$. Les permutations qui ne sont pas réalisées s'appellent des motifs interdits. On sait [J.M. Amigó, S. Elizalde and M. Kennel, $\textit{J. Combin. Theory Ser. A}$ 115 (2008), 485―504] que les motifs interdits les plus courts du $\textit{shift}$ avec $N$ symboles ont longueur $N+2$. Dans cet article on donne une caractérisation des permutations réalisées par le $\textit{shift}$ avec $N$ symboles, et on les dénombre selon leur longueur.

scholarly journals WORD COMPLEXITY AND REPETITIONS IN WORDS

2004 ◽  
Vol 15 (01) ◽  
pp. 41-55 ◽  
Author(s):  
LUCIAN ILIE ◽  
SHENG YU ◽  
KAIZHONG ZHANG
Keyword(s):  
Data Compression ◽  
Fast Algorithms ◽  
Complexity Measure ◽  
Combinatorics On Words ◽  
Complexity Measures ◽  
Infinite Words ◽  
Infinite Word ◽  
De Bruijn ◽  
Subword Complexity

With ideas from data compression and combinatorics on words, we introduce a complexity measure for words, called repetition complexity, which quantifies the amount of repetition in a word. The repetition complexity of w, R (w), is defined as the smallest amount of space needed to store w when reduced by repeatedly applying the following procedure: n consecutive occurrences uu…u of the same subword u of w are stored as (u,n). The repetition complexity has interesting relations with well-known complexity measures, such as subword complexity, SUB , and Lempel-Ziv complexity, LZ . We have always R (w)≥ LZ (w) and could even be that the former is linear while the latter is only logarithmic; e.g., this happens for prefixes of certain infinite words obtained by iterated morphisms. An infinite word α being ultimately periodic is equivalent to: (i) [Formula: see text], (ii) [Formula: see text], and (iii) [Formula: see text]. De Bruijn words, well known for their high subword complexity, are shown to have almost highest repetition complexity; the precise complexity remains open. R (w) can be computed in time [Formula: see text] and it is open, and probably very difficult, to find fast algorithms.

scholarly journals AVOIDING APPROXIMATE SQUARES

2008 ◽  
Vol 19 (03) ◽  
pp. 633-648 ◽  
Author(s):  
PASCAL OCHEM ◽  
NARAD RAMPERSAD ◽  
JEFFREY SHALLIT
Keyword(s):  
Letter Alphabet ◽  
Infinite Word

As is well-known, Axel Thue constructed an infinite word over a 3-letter alphabet that contains no squares, that is, no nonempty subwords of the form xx. In this paper we consider a variation on this problem, where we try to avoid approximate squares, that is, subwords of the form xx' where |x| = |x'| and x and x' are "nearly" identical.

scholarly journals Further Applications of a Power Series Method for Pattern Avoidance

10.37236/621 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Narad Rampersad
Keyword(s):  
Power Series ◽  
Wide Class ◽  
Pattern Avoidance ◽  
Combinatorics On Words ◽  
Factor X ◽  
Series Method ◽  
Power Series Method ◽  
Letter Alphabet ◽  
Binary Alphabet ◽  
Algebraic Technique

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $x$ of $w$ and no non-erasing morphism $h$ from $\Delta^*$ to $\Sigma^*$ such that $h(p) = x$. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a certain wide class of patterns $p$ there are exponentially many words of length $n$ over a $4$-letter alphabet that avoid $p$. We consider some further consequences of their work. In particular, we show that any pattern with $k$ variables of length at least $4^k$ is avoidable on the binary alphabet. This improves an earlier bound due to Cassaigne and Roth.

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