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Partial Words and the Critical Factorization Theorem
 J. Combin. Theory Ser. A
, 2007
"... The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial word ..."
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Cited by 10 (6 self)
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The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial words that are sequences that may have a number of “do not know ” symbols also called “holes”. This paper is devoted to a fundamental result on periods of words, the Critical Factorization Theorem, which states that the period of a word is always locally detectable in at least one position of the word resulting in a corresponding critical factorization. Here, we describe precisely the class of partial words w with one hole for which the weak period is locally detectable in at least one position of w. Our proof provides an algorithm which computes a critical factorization when one exists. A World Wide Web server interface at
Border correlation of binary words
 J. Combin. Theory Ser. A
"... The border correlation function β: A ∗ → A ∗ , for A = {a, b}, specifies which conjugates (cyclic shifts) of a given word w of length n are bordered, in other words, β(w) = c0c1... cn−1, where ci = a or b according to whether the ith cyclic shift σ i (w) of w is unbordered or bordered. Except for ..."
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Cited by 1 (1 self)
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The border correlation function β: A ∗ → A ∗ , for A = {a, b}, specifies which conjugates (cyclic shifts) of a given word w of length n are bordered, in other words, β(w) = c0c1... cn−1, where ci = a or b according to whether the ith cyclic shift σ i (w) of w is unbordered or bordered. Except for some special cases, no binary word w has two consecutive unbordered conjugates (σ i (w) and σ i+1 (w)). We show that this is optimal: in every cyclically overlapfree word every other conjugate is unbordered. We also study the relationship between unbordered conjugates and critical points, as well as, the dynamic system given by iterating the function β. We prove that, for each word w of length n, the sequence w, β(w), β 2 (w),... terminates either in b n or in the cycle of conjugates of the word ab k ab k+1 for n = 2k + 3.
No 714, October 2005On Unique Factorizations of Primitive Words
"... We give a short proof of a result by C. M. Weinbaum [Proc. AMS, 109(3):615– 619, 1990] stating that each a primitive word of length at least 2 has a conjugate w ′ = uv such that both u and v have a unique position in the cyclic word of w. ..."
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We give a short proof of a result by C. M. Weinbaum [Proc. AMS, 109(3):615– 619, 1990] stating that each a primitive word of length at least 2 has a conjugate w ′ = uv such that both u and v have a unique position in the cyclic word of w.
Acta Informatica manuscript No. (will be inserted by the editor)
"... Abstract Let π(w) denote the minimum period of the word w. Let w be a primitive word with period π(w) < w, and z a prefix of w. It is shown that if π(wz) = π(w), then z  < π(w)−gcd(w,z). Detailed improvements of this result are also proven. As a corollary we give a short proof of the fact th ..."
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Abstract Let π(w) denote the minimum period of the word w. Let w be a primitive word with period π(w) < w, and z a prefix of w. It is shown that if π(wz) = π(w), then z  < π(w)−gcd(w,z). Detailed improvements of this result are also proven. As a corollary we give a short proof of the fact that if u,v,w are primitive words such that u 2 is a prefix of v 2, and v 2 is a prefix of w 2, then w > 2u. Finally, we show that each primitive word w has a conjugate w ′ = vu, where w = uv, such that π(w ′ ) = w ′  and u  < π(w). 1
Miscellaneous
, 2003
"... The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. We call a word bordered, if it has a proper prefix which is also a suffix of that word. Let µ(w) denote the maximum length of all unbord ..."
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The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. We call a word bordered, if it has a proper prefix which is also a suffix of that word. Let µ(w) denote the maximum length of all unbordered factors of w, and let ∂(w) denote the period of w. Clearly, µ(w) ≤ ∂(w). We establish that µ(w) = ∂(w), if w has an unbordered prefix of length µ(w) and n ≥ 2µ(w) − 1. This bound is tight and solves the stronger version of a 21 years old conjecture by Duval. It follows from this result that, in general, n ≥ 3µ(w) − 2 implies µ(w) = ∂(w) which gives an improved bound for the question asked by Ehrenfeucht and Silberger in 1979.