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Palindrome complexity
 To appear, Theoret. Comput. Sci
, 2002
"... We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points o ..."
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Cited by 6 (2 self)
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We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to “class P ” of HofKnillSimon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block)complexity. 1
The Subword Complexity of a TwoParameter Family of Sequences
"... We determine the subword complexity of the characteristic functions of a twoparameter family fA n g 1 n=1 of infinite sequences which are associated with the winning strategies for a family of 2player games. A special case of the family has the form A n = bnffc for all n 2 Z?0 , where ff is a f ..."
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Cited by 4 (4 self)
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We determine the subword complexity of the characteristic functions of a twoparameter family fA n g 1 n=1 of infinite sequences which are associated with the winning strategies for a family of 2player games. A special case of the family has the form A n = bnffc for all n 2 Z?0 , where ff is a fixed positive irrational number. The characteristic functions of such sequences have been shown to have subword complexity n + 1. We show that every sequence in the extended family has subword complexity O(n). 1 Introduction Denote by Z0 and Z?0 the set of nonnegative integers and positive integers respectively. Given two heaps of finitely many tokens, we define a 2player heap game as follows. There are two types of moves: 1. Remove any positive number of tokens from a single heap. 2. Remove k ? 0 tokens from one heap and l ? 0 from the other. Here k and l are constrained by the condition: 0 ! k l ! sk + t, where s and t are predetermined positive integers. The player who reaches a stat...
Quasiperiodic and Lyndon episturmian words
 Theoretical Computer Science
"... Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is nonquasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods o ..."
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Cited by 4 (3 self)
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Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is nonquasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their directive words. Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are nonquasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular normalized directive words, which we introduced in an earlier paper. Also of importance is the use of return words to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.
Balanced words
"... Abstract: This article presents a survey about balanced words. The balance property comes from combinatorics on words and is used as a characteristic property of the wellknown Sturmian words. The main goal of this survey is to study various generalizations of this notion with applications and with ..."
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Cited by 2 (0 self)
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Abstract: This article presents a survey about balanced words. The balance property comes from combinatorics on words and is used as a characteristic property of the wellknown Sturmian words. The main goal of this survey is to study various generalizations of this notion with applications and with open problems in number theory and in theoretical computer science. We also prove a new result about the generalized balance property of hypercubic billiard words.
On Correlation Polynomials and Subword Complexity
"... We consider words with letters from a qary alphabet A. The kth subword complexity of a word w ∈ A ∗ is the number of distinct subwords of length k that appear as contiguous subwords of w. We analyze subword complexity from both combinatorial and probabilistic viewpoints. Our first main result is a ..."
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We consider words with letters from a qary alphabet A. The kth subword complexity of a word w ∈ A ∗ is the number of distinct subwords of length k that appear as contiguous subwords of w. We analyze subword complexity from both combinatorial and probabilistic viewpoints. Our first main result is a precise analysis of the expected kth subword complexity of a randomlychosen word w ∈ A n. Our other main result describes, for w ∈ A ∗ , the degree to which one understands the set of all subwords of w, provided that one knows only the set of all subwords of some particular length k. Our methods rely upon a precise characterization of overlaps between words of length k. We use three kinds of correlation polynomials of words of length k: unweighted correlation polynomials; correlation polynomials associated to a Bernoulli source; and generalized multivariate correlation polynomials. We survey previouslyknown results about such polynomials, and we also present some new results concerning correlation polynomials.
SEQUENCES OF LOW ARITHMETICAL COMPLEXITY
 THEORETICAL INFORMATICS AND APPLICATIONS
, 2005
"... Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity. ..."
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Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity.