Results 1  10
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19
The ubiquitous ProuhetThueMorse sequence
 Sequences and their applications, Proceedings of SETA’98
, 1999
"... We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be som ..."
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Cited by 59 (9 self)
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We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.
Permutations avoiding an increasing number of lengthincreasing forbidden subsequences
 Discrete Math. Theor. Comput. Sci
, 2000
"... A permutation is said to ¡ be –avoiding if it does not contain any subsequence having all the same pairwise comparisons ¡ as. This paper concerns the characterization and enumeration of permutations which avoid a set ¢¤ £ of subsequences increasing both in number and in length at the same time. Le ..."
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Cited by 23 (1 self)
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A permutation is said to ¡ be –avoiding if it does not contain any subsequence having all the same pairwise comparisons ¡ as. This paper concerns the characterization and enumeration of permutations which avoid a set ¢¤ £ of subsequences increasing both in number and in length at the same time. Let ¢ £ be the set of subsequences of the “¥§¦©¨�������¦©¨����� � form ¥ ”, being any permutation ��������������¨� � on. ¨��� � For the only subsequence in ¢�� ���� � is and ���� � the –avoiding permutations are enumerated by the Catalan numbers; ¨��� � for the subsequences in ¢� � are, ������ � and the (������������������ � –avoiding permutations are enumerated by the Schröder numbers; for each other value ¨ of greater � than the subsequences in ¢ £ ¨� � are and their length ¦©¨����� � is; the permutations avoiding ¨�� these subsequences are enumerated by a number ������ � �� � � sequence such �������������� � that �� � , being � the –th Catalan number. For ¨ each we determine the generating function of permutations avoiding the subsequences in ¢� £ , according to the length, to the number of left minima and of noninversions.
On the Density of Critical Factorizations
 Theor. Inform. Appl
, 2001
"... We investigate the density of critical positions, that is, the ratio between the number of critical positions and the number of all positions of a word, in in nite sequences of words of index one, that is, the period of which is longer than half of the length of the word. ..."
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Cited by 10 (9 self)
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We investigate the density of critical positions, that is, the ratio between the number of critical positions and the number of all positions of a word, in in nite sequences of words of index one, that is, the period of which is longer than half of the length of the word.
Combinatorial properties of smooth infinite words Srecko Brlek
"... We describe some combinatorial properties of an intriguing class of infinite words connected with the one defined by Kolakoski, defined as the fixed point of the runlength encoding #. It is based on a bijection on the free monoid over # = {1, 2}, that shows some surprising mixing properties. All wo ..."
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Cited by 6 (3 self)
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We describe some combinatorial properties of an intriguing class of infinite words connected with the one defined by Kolakoski, defined as the fixed point of the runlength encoding #. It is based on a bijection on the free monoid over # = {1, 2}, that shows some surprising mixing properties. All words contain the same finite number of square factors, and consequently they are cubefree. This suggests that they have the same complexity as confirmed by extensive computations. We further investigate the occurrences of palindromic subwords. Finally we show that there exist smooth words obtained as fixed points of substitutions (realized by transducers) as in the case of K. 1
Simultaneous avoidance of large squares and fractional powers in infinite binary words
 Int. J. Found. Comput. Sci
, 2004
"... In 1976, Dekking showed that there exists an infinite binary word that contains neither squares yy with y  ≥ 4 nor cubes xxx. We show that ‘cube ’ can be replaced by any fractional power> 5/2. We also consider the analogous problem where ‘4 ’ is replaced by any integer. This results in an interes ..."
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Cited by 5 (0 self)
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In 1976, Dekking showed that there exists an infinite binary word that contains neither squares yy with y  ≥ 4 nor cubes xxx. We show that ‘cube ’ can be replaced by any fractional power> 5/2. We also consider the analogous problem where ‘4 ’ is replaced by any integer. This results in an interesting and subtle hierarchy. 1
Pseudopower Avoidance
, 2012
"... Repetition avoidance has been intensely studied since Thue’s work in the early 1900’s. In this paper, we consider another type of repetition, called pseudopower, inspired by the WatsonCrick complementarity property of DNA sequences. A DNA single strand can be viewed as a string over the fourletter ..."
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Cited by 3 (2 self)
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Repetition avoidance has been intensely studied since Thue’s work in the early 1900’s. In this paper, we consider another type of repetition, called pseudopower, inspired by the WatsonCrick complementarity property of DNA sequences. A DNA single strand can be viewed as a string over the fourletter alphabet{A,C,G,T}, wherein A is the complement of T, while C is the complement of G. Such a DNA single strand will bind to a reverse complement DNA single strand, called its WatsonCrick complement, to form a helical doublestranded DNA molecule. The WatsonCrick complement of a DNA strand is deducible from, and thus informationally equivalent to, the original strand. We use this fact to generalize the notion of the power of a word by relaxing the meaning of “sameness ” to include the image through an antimorphic involution, the model of DNA WatsonCrick complementarity. Given a finite alphabet Σ, an antimorphic involution is a function θ: Σ ∗ −→ Σ ∗ which is an involution, i.e.,θ 2 equals the identity, and an antimorphism, i.e., θ(uv) = θ(v)θ(u), for all u ∈ Σ ∗. For a positive integer k, we call a word w a pseudokthpower with respect to θ if it can be written as w = u1...uk, where for 1 ≤ i,j ≤ k we have either ui = uj or ui = θ(uj). The classical kthpower of a word is a special case of a pseudokthpower, where all the repeating units are identical. We first classify the alphabets Σ and the antimorphic involutions θ for which
Lyndon factorization of the ThueMorse word and its relatives
, 1997
"... this paper, we concentrate on the ThueMorse word and give the computation of its Lyndon factorization (Theorem 3.1) and describe some of its properties (Corollary 3.2, Remark 3.3 and Corollary 3.4). Incidentally, we are able to compute the factorization for the `dual' ThueMorse word in which app ..."
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Cited by 1 (0 self)
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this paper, we concentrate on the ThueMorse word and give the computation of its Lyndon factorization (Theorem 3.1) and describe some of its properties (Corollary 3.2, Remark 3.3 and Corollary 3.4). Incidentally, we are able to compute the factorization for the `dual' ThueMorse word in which appears an infinite Lyndon word (cf Theorem 3.7). We also look at relatives (Equations (4) and (6)) of the ThueMorse word from the same point of view; these were first studied in [7] and [4], and later in [1]. The factorizations given here for these infinite words (cf Theorems 4.6 and 4.7) use morphisms having special properties with respect to Lyndon words. Moreover, we give identities involving these morphisms for these infinite words. 2 Basic Results and Notations
The critical exponent of the Arshon words
, 2008
"... Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n ≥ 2, the critical exponent of the Arshon word of order n is given by (3n − 2)/(2n − 2), and this exponent is attained at position 1. 1 ..."
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Cited by 1 (0 self)
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Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n ≥ 2, the critical exponent of the Arshon word of order n is given by (3n − 2)/(2n − 2), and this exponent is attained at position 1. 1
NONREPETITIVE COLORINGS OF TREES
"... A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by π(G). A famous theorem of Thue asserts that π(P) = 3 for any path P with at least 4 vertices. In ..."
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A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by π(G). A famous theorem of Thue asserts that π(P) = 3 for any path P with at least 4 vertices. In this paper we study the Thue chromatic number of trees. In view of the fact that π(T) is bounded by 4 in this class we aim to describe the 4chromatic trees. In particular, we study the 4critical trees which are minimal with respect to this property. Though there are many trees T with π(T) = 4 we show that any of them has a sufficiently large subdivision H such that π(H) = 3. The proof relies on Thue sequences with additional properties involving palindromic words. We also investigate nonrepetitive edge colorings of trees. By a similar argument we prove that any tree has a subdivision which can be edgecolored by at most ∆ + 1 colors without repetitions on paths.