Abstract
A square is a word of the form uu, where u is a finite word. The problem of determining the number of distinct squares in a finite word was initially explored by Fraenkel and Simpson in 1998. They proved that the number of distinct squares, denoted as \(\textrm{Sq}\left( w\right) \), in a finite word w of length n is upper bounded by 2n and conjectured that \(\textrm{Sq}\left( w\right) \) is no larger than n. In this note, we review some old and new findings concerning the square-counting problem and prove that \(\textrm{Sq}\left( w\right) \le n-\varTheta (\log _2(n))\).
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Notes
- 1.
A Lyndon word indeed.
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Brlek, S., Li, S. (2023). On the Number of Distinct Squares in Finite Sequences: Some Old and New Results. In: Frid, A., Mercaş, R. (eds) Combinatorics on Words. WORDS 2023. Lecture Notes in Computer Science, vol 13899. Springer, Cham. https://doi.org/10.1007/978-3-031-33180-0_3
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