no. 1
On the number of squares in partial words
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 1, pp. 125-138.

The theorem of Fraenkel and Simpson states that the maximum number of distinct squares that a word w of length n can contain is less than 2n. This is based on the fact that no more than two squares can have their last occurrences starting at the same position. In this paper we show that the maximum number of the last occurrences of squares per position in a partial word containing one hole is 2k, where k is the size of the alphabet. Moreover, we prove that the number of distinct squares in a partial word with one hole and of length n is less than 4n, regardless of the size of the alphabet. For binary partial words, this upper bound can be reduced to 3n.

DOI : 10.1051/ita/2010008
Classification : 68R15
Mots clés : square, partial word, theorem of Fraenkel and Simpson 
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     author = {Halava, Vesa and Harju, Tero and K\"arki, Tomi},
     title = {On the number of squares in partial words},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {125--138},
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     zbl = {1184.68372},
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Halava, Vesa; Harju, Tero; Kärki, Tomi. On the number of squares in partial words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 1, pp. 125-138. doi : 10.1051/ita/2010008. http://archive.numdam.org/articles/10.1051/ita/2010008/

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