A well known result of Fraenkel and Simpson states that the number of distinct squares in a word of length is bounded by since at each position there are at most two distinct squares whose last occurrence starts. In this paper, we investigate squares in partial words with one hole, or sequences over a finite alphabet that have a “do not know” symbol or “hole”. A square in a partial word over a given alphabet has the form where is compatible with , and consequently, such square is compatible with a number of words over the alphabet that are squares. Recently, it was shown that for partial words with one hole, there may be more than two squares that have their last occurrence starting at the same position. Here, we prove that if such is the case, then the length of the shortest square is at most half the length of the third shortest square. As a result, we show that the number of distinct squares compatible with factors of a partial word with one hole of length is bounded by .
@article{ITA_2009__43_4_767_0,
author = {Blanchet-Sadri, Francine and Merca\c s, Robert},
title = {A note on the number of squares in a partial word with one hole},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
volume = {43},
year = {2009},
pages = {767-774},
doi = {10.1051/ita/2009019},
mrnumber = {2589991},
zbl = {pre05650346},
language = {en},
url = {http://dml.mathdoc.fr/item/ITA_2009__43_4_767_0}
}
Blanchet-Sadri, Francine; Mercaş, Robert. A note on the number of squares in a partial word with one hole. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) pp. 767-774. doi : 10.1051/ita/2009019. http://gdmltest.u-ga.fr/item/ITA_2009__43_4_767_0/
[1] and , Partial words and a theorem of Fine and Wilf. Theoret. Comput. Sci. 218 (1999) 135-141. | MR 1687780 | Zbl 0916.68120
[2] , Algorithmic Combinatorics on Partial Words. Chapman & Hall/CRC Press, Boca Raton, FL (2008). | MR 2384993 | Zbl pre05199756
[3] and , Conjugacy on partial words. Theoret. Comput. Sci. 289 (2002) 297-312. | MR 1932900 | Zbl 1061.68123
[4] , and , Counting distinct squares in partial words, edited by E. Csuhaj-Varju, Z. Esik, AFL 2008, 12th International Conference on Automata and Formal Languages, Balatonfüred, Hungary (2008) 122-133, www.uncg.edu/cmp/research/freeness | MR 2584159
[5] , and , Equations on partial words. RAIRO-Theor. Inf. Appl. 43 (2009) 23-39, www.uncg.edu/cmp/research/equations | Numdam | MR 2483443 | Zbl 1170.68032
[6] and , How many squares can a string contain? J. Combin. Theory Ser. A 82 (1998) 112-120. | MR 1616571 | Zbl 0910.05001
[7] , A simple proof that a word of length has at most distinct squares. J. Combin. Theory Ser. A 112 (2005) 163-164. | MR 2167480 | Zbl 1088.68146
[8] , A note on the number of squares in a word. Theoret. Comput. Sci. 380 (2007) 373-376. | MR 2331005 | Zbl 1119.68141