Distinct Squares in Circular Words
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چکیده
A circular word, or a necklace, is an equivalence class under conjugation of a word. A fundamental question concerning regularities in standard words is bounding the number of distinct squares in a word of length n. The famous conjecture attributed to Fraenkel and Simpson is that there are at most n such distinct squares, yet the best known upper bound is 1.84n by Deza et al. [Discr. Appl. Math. 180, 52-69 (2015)]. We consider a natural generalization of this question to circular words: how many distinct squares can there be in all cyclic rotations of a word of length n? We prove an upper bound of 3.14n. This is complemented with an infinite family of words implying a lower bound of 1.25n.
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تاریخ انتشار 2017