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 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 Automaticity of primitive words and irreducible polynomials

2013 ◽  
Vol Vol. 15 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Anne Lacroix ◽  
Narad Rampersad
Keyword(s):  
Finite Field ◽  
Finite Automaton ◽  
Prime Numbers ◽  
Deterministic Finite Automaton ◽  
Irreducible Polynomials ◽  
Letter Alphabet ◽  
International Audience ◽  
Primitive Words

Automata, Logic and Semantics International audience If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.

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 On the Number of Balanced Words of Given Length and Height over a Two-Letter Alphabet

2010 ◽  
Vol Vol. 12 no. 3 ◽  
Author(s):  
Nicolas Bédaride ◽  
Eric Domenjoud ◽  
Damien Jamet ◽  
Jean-Luc Rémy
Keyword(s):  
Asymptotic Behaviour ◽  
Generating Functions ◽  
Line Segments ◽  
Integer Points ◽  
Letter Alphabet ◽  
Discrete Line ◽  
International Audience

International audience We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.

scholarly journals On the asymptotic enumeration of accessible automata

2010 ◽  
Vol Vol. 12 no. 3 (Automata, Logic and Semantics) ◽  
Author(s):  
Elcio Lebensztayn
Keyword(s):  
Asymptotic Estimate ◽  
Asymptotic Enumeration ◽  
Lagrange Inversion ◽  
Binomial Series ◽  
Letter Alphabet ◽  
International Audience ◽  
Generalized Binomial Series

Automata, Logic and Semantics International audience We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.

scholarly journals The combinatorics of CAT(0) cubical complexes

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Tia Baker ◽  
Rika Yatchak
Keyword(s):  
Remote Control ◽  
Shortest Path ◽  
Robotic Arm ◽  
General Strategy ◽  
Cubical Complex ◽  
Cubical Complexes ◽  
Square Grid ◽  
Reconfigurable System ◽  
International Audience ◽  
Simultaneous Moves

International audience Given a reconfigurable system $X$, such as a robot moving on a grid or a set of particles traversing a graph without colliding, the possible positions of $X$ naturally form a cubical complex $\mathcal{S}(X)$. When $\mathcal{S}(X)$ is a CAT(0) space, we can explicitly construct the shortest path between any two points, for any of the four most natural metrics: distance, time, number of moves, and number of steps of simultaneous moves. CAT(0) cubical complexes are in correspondence with posets with inconsistent pairs (PIPs), so we can prove that a state complex $\mathcal{S}(X)$ is CAT(0) by identifying the corresponding PIP. We illustrate this very general strategy with one known and one new example: Abrams and Ghrist's ``positive robotic arm" on a square grid, and the robotic arm in a strip. We then use the PIP as a combinatorial ``remote control" to move these robots efficiently from one position to another.

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 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 Submaximal factorizations of a Coxeter element in complex reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vivien Ripoll
Keyword(s):  
Reflection Group ◽  
Coxeter Element ◽  
Partition Lattice ◽  
Combinatorial Object ◽  
Complex Reflection Groups ◽  
Noncrossing Partition ◽  
Finite Reflection Group ◽  
International Audience ◽  
Cas A ◽  
Noncrossing Partition Lattice

International audience When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$. Lorsque $W$ est un groupe de réflexion fini, le treillis $NC(W)$ des partitions non-croisées de type $W$ est un objet combinatoire très riche, qui généralise la notion de partitions non-croisées d'un $n$-gone. Une formule (seulement prouvée au cas par cas à l'heure actuelle) exprime le nombre de chaînes de longueur donnée dans $NC(W)$ sous la forme d'un nombre de Fuß-Catalan généralisé, qui dépend des degrés invariants de $W$. Nous décrivons une stratégie visant à comprendre certaines spécifications de cette formule de manière uniforme, en utilisant une interprétation des chaînes de $NC(W)$ comme fibres d'un "revêtement de Lyashko-Looijenga''. Ce revêtement est construit à partir de la géométrie de l'hypersurface du discriminant de $W$. Nous en déduisons de nouvelles formules de comptage pour certaines factorisations d'un élément de Coxeter de $W$.

scholarly journals Two bijections on Tamari Intervals

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Frédéric Chapoton ◽  
Gregory Chatel ◽  
Viviane Pons
Keyword(s):  
Combinatorial Proof ◽  
Nous Obtenons ◽  
Combinatorial Object ◽  
Combinatorial Objects ◽  
Tamari Lattice ◽  
Initial Rise ◽  
International Audience ◽  
Nous Utilisons

International audience We use a recently introduced combinatorial object, the $\textit{interval-poset}$, to describe two bijections on intervals of the Tamari lattice. Both bijections give a combinatorial proof of some previously known results. The first one is an inner bijection between Tamari intervals that exchanges the $\textit{initial rise}$ and $\textit{lower contacts}$ statistics. Those were introduced by Bousquet-Mélou, Fusy, and Préville-Ratelle who proved they were symmetrically distributed but had no combinatorial explanation. The second bijection sends a Tamari interval to a closed flow of an ordered forest. These combinatorial objects were studied by Chapoton in the context of the Pre-Lie operad and the connection with the Tamari order was still unclear. Nous utilisons les $\textit{intervalles-posets}$, très récemment introduits, pour décrire deux bijections sur les intervalles du treillis de Tamari. Nous obtenons ainsi des preuves combinatoires de précédents résultats. La première bijection est une opération interne sur les intervalles qui échange les statistiques de la $\textit{montée initiale}$ et du $\textit{nombre de contacts}$. Ces dernières ont été introduites par Bousquet-Mélou, Fusy et Préville-Ratelle qui ont prouvé qu’elles étaient symétriquement distribuées sans pour autant proposer d’explication combinatoire. La seconde bijection fait le lien avec un objet étudié par Chapoton dans le cadre de l’opérade Pré-Lie : les flots sur les forêts ordonnées. Le lien avec l’ordre de Tamari avait déjà été remarqué sans pour autant être expliqué.

Sign in / Sign up

Export Citation Format

Share Document