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Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
Unavoidable Binary Patterns
 Acta Informatica
, 1993
"... Peter Roth proved that there are no binary patterns of length six or more that are unavoidable on the twoletter alphabet. He gave an almost complete description of unavoidable binary patterns. In this paper we prove one of his conjectures: the pattern ff 2 fi 2 ff is 2avoidable. From this we d ..."
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Peter Roth proved that there are no binary patterns of length six or more that are unavoidable on the twoletter alphabet. He gave an almost complete description of unavoidable binary patterns. In this paper we prove one of his conjectures: the pattern ff 2 fi 2 ff is 2avoidable. From this we deduce the complete classification of unavoidable binary patterns. We also study the concept of avoidability by iterated morphisms and prove that there are a few 2avoidable patterns which are not avoided by any iterated morphism. 1 Introduction The concept of unavoidable pattern was introduced by Bean, Ehrenfeucht & McNulty [2] and independently by Zimin [10]. They gave a characterization of unavoidable patterns, but there is still no characterization of kunavoidable patterns, i.e. patterns that are unavoidable over a kletter alphabet. However, we can try to find all kunavoidable patterns that can be written with a given alphabet. The case of unary patterns (in other terms: powers of a ...
An algorithm to test if a given circular HDOLlanguage avoids a pattern
 in: IFIP World Computer Congress'94
, 1994
"... To prove that a pattern p is avoidable on a given alphabet, one has to construct an infinite language L that avoids p. Usually, L is a DOLlanguage (obtained by iterating a morphism h) or a HDOLlanguage (obtained by coding a DOLlanguage with another morphism g). Our purpose is to find an algorithm ..."
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To prove that a pattern p is avoidable on a given alphabet, one has to construct an infinite language L that avoids p. Usually, L is a DOLlanguage (obtained by iterating a morphism h) or a HDOLlanguage (obtained by coding a DOLlanguage with another morphism g). Our purpose is to find an algorithm to test, given a HDOLsystem G, whether the language L(G) generated by this system avoids p. We first define the notions of circular morphism, circular DOLsystem and circular HDOLsystem, and we show how to compute the inverse image of a pattern by a circular morphism. Then we prove that by computing successive inverse images of p, we can decide whether the language L(G) avoids p for any fixed pattern p (which may even contain constants), provided that the HDOLsystem G is circular and expansive. 1 Introduction The theory of avoidable patterns, introduced by Zimin [13] and Bean, Ehrenfeucht and McNulty [2], generalizes problems studied by Axel Thue [12] and many others, such as the ex...
On the Entropy and Letter Frequencies of Ternary SquareFree Words
, 2003
"... We enumerate all ternary lengthℓ squarefree words, which are words avoiding squares of words up to length ℓ, for ℓ ≤ 24. We analyse the singular behaviour of the corresponding generating functions. This leads to new upper entropy bounds for ternary squarefree words. We then consider ternary squar ..."
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We enumerate all ternary lengthℓ squarefree words, which are words avoiding squares of words up to length ℓ, for ℓ ≤ 24. We analyse the singular behaviour of the corresponding generating functions. This leads to new upper entropy bounds for ternary squarefree words. We then consider ternary squarefree words with fixed letter densities, thereby proving exponential growth for certain ensembles with various letter densities. We derive consequences for the free energy and entropy of ternary squarefree words.
No iterated morphism generates any Arshon sequence of odd order
 Discr. Math
"... We show that no Arshon sequence of odd order can be generated by an iterated morphism. This solves a problem of Kitaev and generalizes results of Berstel and of Kitaev. Keywords: Combinatorics on words, nonrepetitive words, Arshon’s sequence, DOL systems 1 ..."
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We show that no Arshon sequence of odd order can be generated by an iterated morphism. This solves a problem of Kitaev and generalizes results of Berstel and of Kitaev. Keywords: Combinatorics on words, nonrepetitive words, Arshon’s sequence, DOL systems 1
Words strongly avoiding fractional powers
 Europ. J. of Combinatorics
, 1999
"... Abstract Let k be fixed, 1! k! 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with jxyzj=jxyj * k and where z is a permutation of x. 1 Introduction Nonrepetitive words have been studied since Thue [34]. A word is nonrepetitive if it cannot be writt ..."
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Abstract Let k be fixed, 1! k! 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with jxyzj=jxyj * k and where z is a permutation of x. 1 Introduction Nonrepetitive words have been studied since Thue [34]. A word is nonrepetitive if it cannot be written xyyz, where x; y; z are words, and y is nonempty. Infinite
oro.open.ac.uk On the Entropy and Letter Frequencies of Ternary SquareFree Words
, 2003
"... and other research outputs On the entropy and letter frequencies of ternary squarefree words Journal Article ..."
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and other research outputs On the entropy and letter frequencies of ternary squarefree words Journal Article
Combinatorics on Words: A New Challenging Topic1
, 2004
"... Combinatorics on Words is a relatively new research topic in discrete mathematics, mainly, but not only, motivated by computer science. In this paper we have three major goals. First, starting from very basic definitions we move via some typical proof techniques of the theory to a simply formulated ..."
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Combinatorics on Words is a relatively new research topic in discrete mathematics, mainly, but not only, motivated by computer science. In this paper we have three major goals. First, starting from very basic definitions we move via some typical proof techniques of the theory to a simply formulated open problems. Second, we discuss a few highlights of the theory, and finally we concentrate to connections between combinatorics on words and other branches of mathematics. 1Supported by the Academy of Finland under the grant 44087. Combinatorics on Words (CoW in short) is a relatively new rapidly growing area of discrete mathematics. Its main object is a word, i.e. a sequence – either finite or infinite – of symbols taken from a finite set, usually referred to as an alphabet. The natural environment of a word is that of a free monoid. Consequently, noncommutativity is a fundamental feature of words. Indeed, the words like “no ” and “on ” are not equal.