Circularly Squarefree Words and Unbordered Conjugates: A New Approach

Clokie, Trevor; Gabric, Daniel; Shallit, Jeffrey

doi:10.1007/978-3-030-28796-2_10

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…The Thue-Morse word t is a 2-automatic word, so Walnut is applicable. We wish to keep the discussion on the decision procedure and usage of Walnut brief, so we merely describe the logical formulas necessary to encode our problem and refer the reader to [8] for a proof of Theorem 5.3 using Walnut. See also [34].…”

Section: Theorem 53 If N /mentioning

confidence: 99%

Avoiding abelian powers cyclically

Peltomäki,

Whiteland

2020

Preprint

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We study a new notion of cyclic avoidance of abelian powers. A finite word w avoids abelian N-powers cyclically if for each abelian N-power of period m occurring in the infinite word w ω , we have m ≥ |w|. Let A(k) be the least integer N such that for all n there exists a word of length n over a k-letter alphabet that avoids abelian N-powers cyclically. Let A ∞ (k) be the least integer N such that there exist arbitrarily long words over a k-letter alphabet that avoid abelian N-powers cyclically.We prove that 5 ≤ A(2) ≤ 8, 3 ≤ A(3) ≤ 4, 2 ≤ A(4) ≤ 3, and A(k) = 2 for k ≥ 5. Moreover, we show that A ∞ (2) = 4, A ∞ (3) = 3, and A ∞ (4) = 2.

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Section: Theorem 53 If N /mentioning

confidence: 99%

Avoiding abelian powers cyclically

Peltomäki,

Whiteland

2020

Preprint

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“…In particular, the word u is a factor of the circular word w if and only if u is a factor of some conjugate of w, and the circular word w is square-free if and only if every conjugate of w is square-free. The following theorem was first proven by Currie [5], and has since been reproven by several different methods [6,14].…”

Section: Introductionmentioning

confidence: 99%

Lengths of extremal square-free ternary words

Mol,

Rampersad

2020

Preprint

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A square-free word w over a fixed alphabet Σ is extremal if every word obtained from w by inserting a single letter from Σ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal square-free word, and demonstrated that there are arbitrarily long extremal square-free ternary words. We find all lengths which admit an extremal square-free ternary word. In particular, we show that there is an extremal square-free ternary word of every sufficiently large length. We also solve the analogous problem for circular words.

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“…Several other proofs [21,4,15] of this theorem have now been given, signaling increasing interest in circular words. (See also [6,16].)…”

Section: Introductionmentioning

confidence: 99%