Recently, Constantinescu and Ilie proved a variant of the well-known periodicity theorem of Fine and Wilf in the case of two relatively prime abelian periods and conjectured a result for the case of two non-relatively prime abelian periods. In this paper, we answer some open problems they suggested. We show that their conjecture is false but we give bounds, that depend on the two abelian periods, such that the conjecture is true for all words having length at least those bounds and show that some of them are optimal. We also extend their study to the context of partial words, giving optimal lengths and describing an algorithm for constructing optimal words.
Mots-clés : combinatorics on words, Fine and Wilf's theorem, partial words, abelian periods, periods, optimal lengths
@article{ITA_2013__47_3_215_0,
author = {Blanchet-Sadri, Francine and Simmons, Sean and Tebbe, Amelia and Veprauskas, Amy},
title = {Abelian periods, partial words, and an extension of a theorem of {Fine} and {Wilf}},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {215--234},
publisher = {EDP-Sciences},
volume = {47},
number = {3},
year = {2013},
doi = {10.1051/ita/2013034},
mrnumber = {3103125},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ita/2013034/}
}
TY - JOUR AU - Blanchet-Sadri, Francine AU - Simmons, Sean AU - Tebbe, Amelia AU - Veprauskas, Amy TI - Abelian periods, partial words, and an extension of a theorem of Fine and Wilf JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2013 SP - 215 EP - 234 VL - 47 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ita/2013034/ DO - 10.1051/ita/2013034 LA - en ID - ITA_2013__47_3_215_0 ER -
%0 Journal Article %A Blanchet-Sadri, Francine %A Simmons, Sean %A Tebbe, Amelia %A Veprauskas, Amy %T Abelian periods, partial words, and an extension of a theorem of Fine and Wilf %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2013 %P 215-234 %V 47 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ita/2013034/ %R 10.1051/ita/2013034 %G en %F ITA_2013__47_3_215_0
Blanchet-Sadri, Francine; Simmons, Sean; Tebbe, Amelia; Veprauskas, Amy. Abelian periods, partial words, and an extension of a theorem of Fine and Wilf. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 47 (2013) no. 3, pp. 215-234. doi : 10.1051/ita/2013034. http://www.numdam.org/articles/10.1051/ita/2013034/
[1] , , and , On shortest crucial words avoiding abelian powers. Discrete Appl. Math. 158 (2010) 605-607. | MR | Zbl
[2] , and , On abelian versions of the critical factorization theorem. In JM 2010, 13ièmes Journées Montoises d'Informatique Théorique, Amiens, France (2010). | Numdam | Zbl
[3] and , Partial words and a theorem of Fine and Wilf. Theoret. Comput. Sci. 218 (1999) 135-141. | MR | Zbl
[4] , Algorithmic Combinatorics on Partial Words. Chapman & Hall/CRC Press, Boca Raton, FL (2008). | MR | Zbl
[5] , , , , and , Avoiding abelian squares in partial words. J. Combin. Theory Ser. A 119 (2012) 257-270. | MR | Zbl
[6] , and , Periods in partial words: An algorithm. J. Discrete Algorithms 16 (2012) 113-128. | MR | Zbl
[7] , and , Fine and Wilf's theorem for partial words with arbitrarily many weak periods. Internat. J. Foundations Comput. Sci. 21 (2010) 705-722. | MR | Zbl
[8] , and , Abelian repetitions in partial words. Adv. Appl. Math. 48 (2012) 194-214. | MR | Zbl
[9] , and , Fine and Wilf's theorem for abelian periods in partial words. In JM 2010, 13ièmes Journées Montoises d'Informatique Théorique, Amiens, France (2010).
[10] , and , Fine and Wilf's theorem for three periods and a generalization of Sturmian words. Theoret. Comput. Sci. 218 (1999) 83-94. | MR | Zbl
[11] and , Combinatorics of Words. In Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa, Springer-Verlag, Berlin Vol. 1 (1997) 329-438. | MR | Zbl
[12] and , Generalised Fine and Wilf's theorem for arbitrary number of periods. Theor. Comput. Sci. 339 (2005) 49-60. | MR | Zbl
[13] and , Fine and Wilf's theorem for abelian periods. Bull. Eur. Assoc. Theor. Comput. Sci. 89 (2006) 167-170. | MR | Zbl
[14] and , Weak repetitions in strings. J. Combin. Math. Combin. Comput. 24 (1997) 33-48. | MR | Zbl
[15] and , A cyclic binary morphism avoiding abelian fourth powers. Theoret. Comput. Sci. 410 (2009) 44-52. | MR | Zbl
[16] and , Abelian primitive words. In DLT 2011, 15th International Conference on Developments in Language Theory, Milano, Italy, Lect. Notes Comput. Sci. Vol. 6795 edited by G. Mauri and A. Leporati. Springer-Verlag, Berlin, Heidelberg (2011) 204-215. | MR | Zbl
[17] , , and , Computing abelian periods in words. PSC 2011, Prague Stringology Conference, Prague, Czech Republic, (2011) 184-196.
[18] and , Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109-114. | MR | Zbl
[19] , and , Interaction properties of relational periods. Discrete Math. Theoret. Comput. Sci. 10 (2008) 87-112. | MR | Zbl
[20] , On a paper by Castelli, Mignosi, Restivo. Theoret. Inform. Appl. 34 (2000) 373-377. | Numdam | MR | Zbl
[21] , Abelian squares are avoidable on 4 letters. In ICALP 1992, 19th International Colloquium on Automata, Languages and Programming, Lect. Notes Comput. Sci. Vol. 623 edited by W. Kuich. Springer-Verlag, Berlin (1992) 241-52. | MR
[22] and , On abelian repetition threshold. In JM 2010, 13ièmes Journées Montoises d'Informatique Théorique, Amiens, France (2010). | Numdam | Zbl
[23] and , Partial words and the interaction property of periods. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 68 (2004) 191-214. | MR | Zbl
[24] and , On the periods of partial words. In MFCS 2001, 26th International Symposium on Mathematical Foundations of Computer Science, Lect. Notes Comput. Sci. Vol. 2136 edited by J. Sgall, A. Pultr and P. Kolman. London, UK, Springer-Verlag. (2001) 657-665. | MR | Zbl
[25] and , A new approach to the periodicity lemma on strings with holes. Theoret. Comput. Sci. 410 (2009) 4295-4302. | MR | Zbl
[26] and , Fine and Wilf words for any periods. Indagationes Math. 14 (2003) 135-147. | MR | Zbl
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