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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.
@article{ITA_2010__44_1_125_0,
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},
publisher = {EDP-Sciences},
volume = {44},
number = {1},
year = {2010},
doi = {10.1051/ita/2010008},
mrnumber = {2604938},
zbl = {1184.68372},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ita/2010008/}
}
TY - JOUR AU - Halava, Vesa AU - Harju, Tero AU - Kärki, Tomi TI - On the number of squares in partial words JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2010 SP - 125 EP - 138 VL - 44 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ita/2010008/ DO - 10.1051/ita/2010008 LA - en ID - ITA_2010__44_1_125_0 ER -
%0 Journal Article %A Halava, Vesa %A Harju, Tero %A Kärki, Tomi %T On the number of squares in partial words %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2010 %P 125-138 %V 44 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ita/2010008/ %R 10.1051/ita/2010008 %G en %F ITA_2010__44_1_125_0
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
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