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Transductions and contextfree languages
 Ed. Teubner
, 1979
"... 1.1 Notation and examples......................... 3 ..."
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1.1 Notation and examples......................... 3
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.
Polynomial versus exponential growth in repetitionfree binary words
 To appear, J. Combinatorial Theory Ser. A
, 2003
"... It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of le ..."
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It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of length n that avoid 7 3powers, but there are exponentially many binary words of length n that avoid 7+ 3powers. This answers an open question of Kobayashi from 1986. 1
There are more than 2 n/17 nletter ternary squarefree words
 J. Integer Seq
, 1998
"... Abstract: We prove that the ‘connective constant ’ for ternary squarefree words is at least 2 1/17 = 1.0416..., improving on Brinkhuis and Brandenburg’s lower bounds of 2 1/24 = 1.0293... and 2 1/22 = 1.032... respectively. This is the first improvement since 1983. A word is squarefree if it never ..."
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Abstract: We prove that the ‘connective constant ’ for ternary squarefree words is at least 2 1/17 = 1.0416..., improving on Brinkhuis and Brandenburg’s lower bounds of 2 1/24 = 1.0293... and 2 1/22 = 1.032... respectively. This is the first improvement since 1983. A word is squarefree if it never stutters, i.e. if it cannot be written as axxb for words a,b and nonempty word x. For example, ‘example ’ is squarefree, but ‘exampample ’ is not. See Steven Finch’s famous Mathematical Constants site[3] for a thorough discussion and many references. Let a(n) be the number of ternary squarefree nletter words ( A006156, M2550 in the SloanePlouffe[4] listing, 1,3,6,12,18,30,42,...). Brinkhuis[2] and Brandenburg[1] showed that a(n) ≥ 2 n/24, and a(n) ≥ 2 n/22 respectively. Here we show, by extending the method of [2], that a(n) ≥ 2 n/17, and hence that µ: = limn→ ∞ a(n) 1/n ≥ 2 1/17 = 1.0416.... Definition: A triplepair [[U0,V0], [U1,V1], [U2,V2]] where U0,V0,U1,V1,U2,V2 are words in the alphabet {0,1,2} of the same length k, will be called a kBrinkhuis triplepair if the following conditions are satisfied. • The 24 words of length 2k,
Counting OverlapFree Binary Words
 Springer LNCS 665
, 1993
"... A word on a finite alphabet A is said to be overlapfree if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlapfree binary words of length n, which is known to be bounded by a polynomial in n. First, we describe ..."
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A word on a finite alphabet A is said to be overlapfree if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlapfree binary words of length n, which is known to be bounded by a polynomial in n. First, we describe a bijection between the set of overlapfree words and a rational language. This yields recurrence relations for un , which allow to compute un in logarithmic time. Then, we prove that the numbers ff = sup f r j n r = O (un) g and fi = inf f r j un = O (n r ) g are distinct, and we give an upper bound for ff and a lower bound for fi. Finally, we compute an asymptotically tight bound to the number of overlapfree words of length less than n. 1 Introduction In general, the problem of evaluating the number un of words of length n in the language U consisting of words on some finite alphabet A with no factors in a certain set F is not easy. If F is finite, it amounts to counting words in...
The entropy of squarefree words
 Math. Comput. Modelling
, 1997
"... Finite alphabets of at least three letters permit the construction of squarefree words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid converge ..."
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Finite alphabets of at least three letters permit the construction of squarefree words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters. Résumé Il est possible de construire des mots de longueur infinie sans carré sur un alphabet ayant au moins trois lettres. Nous démontrons que l’entropie du langage des mots sans carré sur un tel alphabet est strictement positive et l’encadrons par des bornes inférieure et supérieure raisonnables. Enfin, nous donnons pour l’entropie une expression approchée qui est asymptotiquement correcte et converge rapidement lorsque le nombre de lettres de l’alphabet tend vers l’infini.
Binary words containing infinitely many overlaps
"... We characterize the squares occurring in infinite overlapfree binary words and construct various α powerfree binary words containing infinitely many overlaps. 1 ..."
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We characterize the squares occurring in infinite overlapfree binary words and construct various α powerfree binary words containing infinitely many overlaps. 1
A characterization of 2+free words over a binary alphabet
, 1995
"... It is shown that 2+repetition, i.e. a word of the form uvuvu where u is a letter and v is a word, is the smallest repetition which can be avoided in infinite words over binary alphabet. Such binary words avoiding pattern uvuvu, finite or infinite, are called as 2+free words and those words are the ..."
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It is shown that 2+repetition, i.e. a word of the form uvuvu where u is a letter and v is a word, is the smallest repetition which can be avoided in infinite words over binary alphabet. Such binary words avoiding pattern uvuvu, finite or infinite, are called as 2+free words and those words are the main topic of this work. It is shown here that 2+free words over binary alphabet can be presented as words built from special kind of blocks, called Morseblocks, with some rules. In particular, the given presentation by these blocks is unique for 2+free words long enough. Moreover, it is also shown that the language generated by this presentation can be described by some automaton. In fact, the corresponding presentation in blocks for finite 2