On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer Science (accepted October 15, 2007

Combinatorial aspects of Escher tilings

International audienceIn 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

A parallelogram tile fills the plane by translation in at most two distinct ways

We consider the tilings by translation of a single polyomino or tile on the square grid Z2 (Z exposant 2). It is well-known that there are two regular tilings of the plane, namely, parallelogram and hexagonal tilings. Although there exist tiles admitting an arbitrary number of distinct hexagon tilings, it has been conjectured that no polyomino admits more than two distinct parallelogram tilings. In this paper, we prove this conjecture

Two infinite families of polyominoes that tile the plane by translation in two distinct ways

It has been proved that, among the polyominoes that tile the plane by translation, the so-called squares tile the plane in at most two distinct ways. In this paper, we focus on double squares, that is, the polyominoes that tile the plane in exactly two distinct ways. Our approach is based on solving equations on words, which allows us to exhibit properties about their shape. Moreover, we describe two infinite families of double squares. The first one is directly linked to Christoffel words and may be interpreted as segments of thick straight lines. The second one stems from the Fibonacci sequence and reveals some fractal features

Smooth Words on a 2-letter alphabets having same parity

International audienceIn this paper, we consider smooth words over 2-letter alphabets {a, b}, where a, b are integers having same parity, with 0 < a < b. We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either a = 1 and b odd, or a, b are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for a = 1 with b odd, the frequency of b's is 1/(√2b − 1 + 1)

Efficient operations on discrete paths

We present linear time and space operations on discrete paths. First, we compute the outer hull of any discrete path. As a consequence, a linear time and space algorithm is obtained for computing the convex hull. Next, we provide a linear algorithm computing the overlay graph of two simple closed paths. From this overlay graph, one can easily compute the intersection, union and difference of two Jordan polyominoes, i.e. polyominoes whose boundary is a Jordan curve. The linear complexity is obtained by using an enriched version of a data structure introduced by Brlek, Koskas and Provençal: a quadtree for representing points in the discrete plane augmented with neighborhood links, which was introduced in particular to decide in linear time if a discrete path is self-intersecting

Sur la complexite des chaines d'operations dans les monoiedes

SIGLECNRS T Bordereau / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Extremal generalized smooth words

International audienceIn this article, we consider smooth words over 2-letter alphabets {a, b}, with a, b 2 N having same parity. We show that they all are recurrent and provide a linear algorithm computing the extremal words. Moreover, the set of factors of any infinite smooth word over an odd alphabet is closed under reversal, while it is not for even parity alphabets. The minimal word is an infinite Lyndon word if and only if either a = 1 and b odd, or a, b even. We also describe a connection between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets. Finally, the density of letters in extremal words is 1/2 for even alphabets, and 1/(p2b − 1 − 1) for a = 1 with b odd

Properties of the extremal infinite smooth words

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Properties of the extremal infinite smooth words

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