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On the number of squares in partial words

Published online by Cambridge University Press:  11 February 2010

Vesa Halava
Affiliation:
Department of Mathematics and Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland; vehalava@utu.fi, harju@utu.fi, topeka@utu.fi
Tero Harju
Affiliation:
Department of Mathematics and Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland; vehalava@utu.fi, harju@utu.fi, topeka@utu.fi
Tomi Kärki
Affiliation:
Department of Mathematics and Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland; vehalava@utu.fi, harju@utu.fi, topeka@utu.fi Institute of Mathematics, University of Liège, Grand Traverse 12 (B 37), 4000 Liège, Belgium.
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Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, 2010

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References

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