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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 ...
FURTHER APPLICATIONS OF A POWER SERIES METHOD FOR PATTERN AVOIDANCE
, 907
"... Abstract. In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet ∆ if there is no factor x of w and no nonerasing morphism h from ∆ ∗ to Σ ∗ such that h(p) = x. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a ..."
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Abstract. In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet ∆ if there is no factor x of w and no nonerasing morphism h from ∆ ∗ to Σ ∗ 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 4letter 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. 1.
Avoidingsufficientlylargebinarypatterns
"... U. Schmidt proved that any binary pattern of length at least 13 is avoided by either the Thue–Morse word or the Fibonacci word. We give a somewhat simpler proof of this result. 1 ..."
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U. Schmidt proved that any binary pattern of length at least 13 is avoided by either the Thue–Morse word or the Fibonacci word. We give a somewhat simpler proof of this result. 1