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 
@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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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. http://www.numdam.org/articles/10.1051/ita/2010008/

[1] J.-P. Allouche and J. Shallit, The ubiquitous Prouhet-Thue-Morse sequence, in Sequences and Their Applications: Proceedings of SETA'98, edited by C. Ding, T. Helleseth and H. Niederreiter. Springer, London (1999) 1-16. | Zbl

[2] J. Berstel and L. Boasson, Partial words and a theorem of Fine and Wilf. Theoret. Comput. Sci. 218 (1999) 135-141. | Zbl

[3] F. Blanchet-Sadri, Algorithmic Combinatorics on Partial Words. Chapman & Hall/CRC Press, Boca Raton, FL (2007). | Zbl

[4] F. Blanchet-Sadri, R. Mercaş and G. Scott, Counting distinct squares in partial words, in Proceedings of the 12th International Conference on Automata and Formal Languages (AFL 2008), edited by E. Csuhaj-Varjú and Z. Ésik. Balatonfüred, Hungary (2008) 122-133. Also available at http://www.uncg.edu/cmp/research/freeness/distinctsquares.pdf | Zbl

[5] M. Crochemore and W. Rytter, Squares, cubes, and time-space efficient string searching. Algorithmica 13 (1995) 405-425. | Zbl

[6] A.S. Fraenkel and J. Simpson, How many squares can a string contain? J. Combin. Theory Ser. A 82 (1998) 112-120. | Zbl

[7] V. Halava, T. Harju, T. Kärki and P. Séébold, Overlap-freeness in infinite partial words. Theoret. Comput. Sci. 410 (2009) 943-948. | Zbl

[8] V. Halava, T. Harju and T. Kärki, Square-free partial words. Inform. Process. Lett. 108 (2008) 290-292. | Zbl

[9] L. Ilie, A simple proof that a word of length n has at most 2n distinct squares. J. Combin. Theory Ser. A 112 (2005) 163-164. | Zbl

[10] L. Ilie, A note on the number of squares in a word. Theoret. Comput. Sci. 380 (2007) 373-376. | Zbl

[11] M. Lothaire, Combinatorics on Words. Encyclopedia of Mathematics 17, Addison-Wesley (1983). | Zbl

[12] M. Lothaire, Algebraic combinatorics on words. Encyclopedia of Mathematics and its Applications 90, Cambridge University Press (2002). | Zbl

[13] F. Manea and R. Mercaş, Freeness of partial words. Theoret. Comput. Sci. 389 (2007) 265-277. | Zbl

[14] A. Thue, Über unendliche Zeichenreihen. Norske Vid. Skrifter I Mat.-Nat. Kl., Christiania 7 (1906) 1-22. | JFM

[15] A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Skrifter I Mat.-Nat. Kl., Christiania 1 (1912) 1-67. | JFM

Cité par Sources :