Extremal Pattern-Avoiding Words

Ter-Saakov, Natalya; Zhang, Emily

doi:10.48550/arxiv.2009.10186

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2021

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“…It is known that there exist infinitely many abelian square-free words over a 4-letter alphabet, as conjectured by Erdős [6] and proved by Keränen [9]. Ter-Saakov and Zhang found in [14] the shortest extremal abelian square-free word over four letters:…”

Section: Final Discussionmentioning

confidence: 99%

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Square-free extensions of words

Grytczuk¹,

Kordulewski²,

Pawlik³

2021

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A word is square-free if it does not contain nonempty factors of the form XX. In 1906 Thue proved that there exist arbitrarily long square-free words over a 3-letter alphabet.It was proved recently [8] that among these words there are infinitely many extremal ones, that is, having a square in every single-letter extension.We study diverse problems concerning extensions of words preserving the property of avoiding squares. Our main motivation is the conjecture stating that there are no extremal words over a 4-letter alphabet. We also investigate a natural recursive procedure of generating square-free words by a single-letter right-most extension. We present the results of computer experiments supporting a supposition that this procedure gives an infinite squarefree word over any alphabet of size at least three.

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Section: Final Discussionmentioning

confidence: 99%

“…A first attempt in studying extremal P -free words was made by Ter-Saakov and Zhang in [14], though they focused on a special family of unavoidable patterns of the form P =…”

Section: Final Discussionmentioning

confidence: 99%