Results 1  10
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21
How Many Squares Must a Binary Sequence Contain?
, 1994
"... . Let g(n) be the length of a longest binary string containing at most n distinct squares (two identical adjacent substrings). Then g(0) = 3 (010 is such a string), g(1) = 7 (0001000) and g(2) = 18 (010011000111001101). How does the sequence \Phi g(n) \Psi behave? We give a complete answer. 1. ..."
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Cited by 21 (7 self)
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. Let g(n) be the length of a longest binary string containing at most n distinct squares (two identical adjacent substrings). Then g(0) = 3 (010 is such a string), g(1) = 7 (0001000) and g(2) = 18 (010011000111001101). How does the sequence \Phi g(n) \Psi behave? We give a complete answer. 1. Introduction A binary word (or string) containing no square (a pair of identical adjacent subwords) has maximum length 3; in fact, the only squarefree words of length 3 are 010 and its 1complement 101. A computer disclosed that a binary word containing at most 1 square has maximum length 7: the only words of length 7 with only 1 square are 0001000; 0100010; 0111011 and their 1complements and the reverse of 0111011 and its 1complement. Further, a binary word containing at most 2 distinct squares has maximum length 18; the only words of length 18 which contain only 2 distinct squares are 010011000111001101 and its 1complement (which is also its reverse). In general, let g(k) denote t...
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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Cited by 12 (5 self)
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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.
Weak Repetitions In Strings
 J. Combinatorial Mathematics and Combinatorial Computing
"... A weak repetition in a string consists of two or more adjacent substrings which are permutations of each other. We describe a straightforward \Theta(n 2 ) algorithm which computes all the weak repetitions in a given string of length n defined on an arbitrary alphabet A. Using results on Fibonacci ..."
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Cited by 11 (1 self)
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A weak repetition in a string consists of two or more adjacent substrings which are permutations of each other. We describe a straightforward \Theta(n 2 ) algorithm which computes all the weak repetitions in a given string of length n defined on an arbitrary alphabet A. Using results on Fibonacci and other simple strings, we prove that this algorithm is asymptotically optimal over all known encodings of the output. 1 INTRODUCTION Interest in the periodic behaviour of strings dates back to Thue [T06] at the turn of the century. Thue considered what we call here strong repetitions (equal adjacent substrings) and showed how to construct an infinitely long string on an alphabet of only three letters with no strong repetitions. (Other constructions on three letters have been discovered several times since, most recently by Dekking [D79] and Pleasants [P70]  the latter lists several references to earlier constructions.) More recently, Erdos [E61, p. 240] considered "Abelian squares" (w...
NONREPETITIVE COLORINGS OF GRAPHS
"... A sequence a = a1a2...an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long no ..."
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Cited by 9 (0 self)
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A sequence a = a1a2...an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long nonrepetitive sequences. In this paper we consider a natural generalization of Thue’s sequences for colorings of graphs. A coloring of the set of edges of a given graph G is nonrepetitive if the sequence of colors on any path in G is nonrepetitive. We call the minimal number of colors needed for such a coloring the Thue number of G and denote it by π(G). The main problem we consider is the relation between the numbers π(G) and ∆(G). We show, by an application of the Lovász Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π(G) ≤ c∆(G) 2 for some absolute constant c. For certain special classes of graphs we obtain linear upper bounds on π(G), by giving explicit colorings. For instance, the Thue number of the complete graph Kn is at most 2n − 3, and π(T) ≤ 4(∆(T) − 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.
ThueLike Sequences and Rainbow Arithmetic Progressions
 ELECTRONIC J. COMBINATORICS
, 2002
"... A sequence u = u 1 u 2 :::u n is said to be nonrepetitive if no two adjacent blocks of u are exactly the same. For instance, the sequence abcbcba contains a repetition bcbc, while abcacbabcbac is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive seq ..."
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Cited by 6 (0 self)
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A sequence u = u 1 u 2 :::u n is said to be nonrepetitive if no two adjacent blocks of u are exactly the same. For instance, the sequence abcbcba contains a repetition bcbc, while abcacbabcbac is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set fa; b; cg. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this
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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Cited by 5 (1 self)
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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.
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
Words avoiding abelian inclusions
 J. Automata, Languages and Combinatorics. OttovonGuerickeUniv
"... ABSTRACT We study a generalization of abelian squares which we call abelian inclusions: a word uv is said to be an f(l)inclusion if the commutative image of v majorizes that of u, and jvj ^ juj + f(juj). We prove that clinclusions are unavoidable, but cinclusions are avoidable for an arbitrary co ..."
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Cited by 1 (0 self)
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ABSTRACT We study a generalization of abelian squares which we call abelian inclusions: a word uv is said to be an f(l)inclusion if the commutative image of v majorizes that of u, and jvj ^ juj + f(juj). We prove that clinclusions are unavoidable, but cinclusions are avoidable for an arbitrary constant c.