A1 Refereed original research article in a scientific journal
ON THE NUMBER OF SQUARES IN PARTIAL WORDS
Authors: Harju Tero, Halava Vesa, Kärki Tomi
Publisher: EDP SCIENCES S A
Publication year: 2010
Journal: RAIRO: Informatique Théorique et Applications / RAIRO: Theoretical Informatics and Applications
Journal name in source: RAIRO-THEORETICAL INFORMATICS AND APPLICATIONS
Journal acronym: RAIRO-THEOR INF APPL
Number in series: 1
Volume: 44
Issue: 1
First page : 125
Last page: 138
Number of pages: 14
ISSN: 0988-3754
DOI: https://doi.org/10.1051/ita/2010008
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.
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.
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