Avoidance bases for formulas with reversal

Currie, James D.; Mol, Lucas; Rampersad, Narad

doi:10.1016/j.tcs.2018.04.039

Cited by 8 publications

(6 citation statements)

References 8 publications

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“…Finally, we remark that the study of k-avoidability of patterns up to ≃ has implications for the k-avoidability of patterns with reversal (see [9,12,13] for definitions and examples). In particular, if the pattern p is k-avoidable up to ≃, then all patterns with reversal that are obtained by swapping any number of letters in p with their mirror images are simultaneously k-avoidable; that is, there is an infinite word on k letters avoiding all such "decorations" of p.…”

Section: Discussionmentioning

confidence: 99%

The undirected repetition threshold and undirected pattern avoidance

Currie¹,

Mól²

2020

Preprint

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For a rational number r such that 1 < r ≤ 2, an undirected r-power is a word of the form xyx ′ , where the word x is nonempty, the word x ′ is in {x, x R }, and we have |xyx ′ |/|xy| = r. The undirected repetition threshold for k letters, denoted URT(k), is the infimum of the set of all r such that undirected r-powers are avoidable on k letters. We first demonstrate that URT(3) = 7 4 . Then we show that URT(k) ≥ k−1 k−2 for all k ≥ 4. We conjecture that URT(k) = k−1 k−2 for all k ≥ 4, and we confirm this conjecture for k ∈ {4, 5, . . . , 21}. We then consider related problems in pattern avoidance; in particular, we find the undirected avoidability index of every binary pattern. This is an extended version of a paper presented at WORDS 2019, and it contains new and improved results. MSC 2010: 68R15

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Section: Discussionmentioning

confidence: 99%

The undirected repetition threshold and undirected pattern avoidance

Currie¹,

Mól²

2020

Preprint

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“…We plan to determine the avoidability index of all binary patterns up to ≃ in a future work. Finally, we remark that the study of k-avoidability of patterns up to ≃ has implications for k-avoidability of patterns with reversal (see [7,10,11] for definitions and examples). In particular, if pattern p is k-avoidable up to ≃, then all patterns with reversal that are obtained by swapping any number of letters in p with their mirror images are simultaneously k-avoidable; that is, there is an infinite word on k letters avoiding all such "decorations" of p.…”

Section: Related Problems In Pattern Avoidancementioning

confidence: 99%

The Undirected Repetition Threshold

Currie

Mol

2019

Lecture Notes in Computer Science

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For rational 1 < r ≤ 2, an undirected r-power is a word of the form xyx ′ , where x is nonempty, x ′ ∈ {x, x R }, and |xyx ′ |/|xy| = r. The undirected repetition threshold for k letters, denoted URT(k), is the infimum of the set of all r such that undirected r-powers are avoidable on k letters. We first demonstrate that URT(3) = 7 4 . Then we show that URT(k) ≥ k−1 k−2 for all k ≥ 4. We conjecture that URT(k) = k−1 k−2 for all k ≥ 4, and we confirm this conjecture for k ∈ {4, 8, 12}.

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“…An occurrence of a formula f in a word w is a non-erasing morphism h such that the h-image of every fragment of f is a factor of w. As for patterns, the avoidability index λ(f ) of a formula f is the size of the smallest alphabet allowing the existence of an infinite word containing no occurrence of f . Recently, the avoidability of formulas with reversal has been considered by Currie, Mol, and Rampersad [2,3] and me [7].…”

Section: Preliminariesmentioning

confidence: 99%

Doubled Patterns are 3-Avoidable

Ochem¹

2016

Electron. J. Combin.

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In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet ∆ if there is no factor f of w such that f = h(p) where h : ∆ * → Σ * is a non-erasing morphism. A pattern p is said to be k-avoidable if there exists an infinite word over a k-letter alphabet that avoids p. A pattern is said to be doubled if no variable occurs only once. Doubled patterns with at most 3 variables and patterns with at least 6 variables are 3-avoidable. We show that doubled patterns with 4 and 5 variables are also 3-avoidable.

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Avoidance bases for formulas with reversal

Cited by 8 publications

References 8 publications

The undirected repetition threshold and undirected pattern avoidance

The undirected repetition threshold and undirected pattern avoidance

The Undirected Repetition Threshold

Doubled Patterns are 3-Avoidable

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