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 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 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.

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 Grasshopper Avoidance of Patterns

10.37236/6210 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Michał Dębski ◽  
Urszula Pastwa ◽  
Krzysztof Węsek
Keyword(s):  
Euclidean Plane ◽  
Pattern Avoidance ◽  
Type Problem ◽  
Compression Method ◽  
New Variant ◽  
Classical Pattern ◽  
Entropy Compression ◽  
Line P ◽  
Entropy Compression Method

Motivated by a geometrical Thue-type problem, we introduce a new variant of the classical pattern avoidance in words, where jumping over a letter in the pattern occurrence is allowed. We say that pattern $p\in E^+$ occurs with jumps in a word $w=a_1a_2\ldots a_k \in A^+$, if there exist a non-erasing morphism $f$ from $E^*$ to $A^*$ and a sequence $(i_1, i_2, \ldots , i_l)$ satisfying $i_{j+1}\in\{ i_j+1, i_j+2 \}$ for $j=1, 2, \ldots, l-1$, such that $f(p) = a_{i_1}a_{i_2}\ldots a_{i_l}.$ For example, a pattern $xx$ occurs with jumps in a word $abdcadbc$ (for $x \mapsto abc$). A pattern $p$ is grasshopper $k$-avoidable if there exists an alphabet $A$ of $k$ elements, such that there exist arbitrarily long words over $A$ in which $p$ does not occur with jumps. The minimal such $k$ is the grasshopper avoidability index of $p$. It appears that this notion is related to two other problems: pattern avoidance on graphs and pattern-free colorings of the Euclidean plane. In particular, we show that a sequence avoiding a pattern $p$ with jumps can be a tool to construct a line $p$-free coloring of $\mathbb{R}^2$.    In our work, we determine the grasshopper avoidability index of patterns $\alpha^n$ for all $n$ except $n=5$. We also show that every doubled pattern is grasshopper $(2^7+1)$-avoidable, every pattern on $k$ variables of length at least $2^k$ is grasshopper $37$-avoidable, and there exists a constant $c$ such that every pattern of length at least $c$ on $2$ variables is grasshopper $3$-avoidable (those results are proved using the entropy compression method).

scholarly journals Gray code order for Lyndon words

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Vincent Vajnovszki
Keyword(s):  
Gray Code ◽  
Time Algorithm ◽  
Positive Answer ◽  
Combinatorics On Words ◽  
International Audience ◽  
Lyndon Words

International audience At the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order.

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].

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