Existence of an infinite ternary 64-abelian square-free word
Huova, Mari
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 48 (2014), p. 307-314 / Harvested from Numdam
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We consider a recently defined notion of k-abelian equivalence of words by concentrating on avoidance problems. The equivalence class of a word depends on the numbers of occurrences of different factors of length k for a fixed natural number k and the prefix of the word. We have shown earlier that over a ternary alphabet k-abelian squares cannot be avoided in pure morphic words for any natural number k. Nevertheless, computational experiments support the conjecture that even 3-abelian squares can be avoided over ternary alphabets. In this paper we establish the first avoidance result showing that by choosing k to be large enough we have an infinite k-abelian square-free word over three letter alphabet. In addition, this word can be obtained as a morphic image of a pure morphic word.

Publié le : 2014-01-01
DOI : https://doi.org/10.1051/ita/2014012
Classification:  68R15 
@article{ITA_2014__48_3_307_0,
     author = {Huova, Mari},
     title = {Existence of an infinite ternary 64-abelian square-free word},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     volume = {48},
     year = {2014},
     pages = {307-314},
     doi = {10.1051/ita/2014012},
     mrnumber = {3302490},
     zbl = {1297.68192},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ITA_2014__48_3_307_0}
}

Huova, Mari. Existence of an infinite ternary 64-abelian square-free word. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 48 (2014) pp. 307-314. doi : 10.1051/ita/2014012. http://gdmltest.u-ga.fr/item/ITA_2014__48_3_307_0/

[1] J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003). | MR 1997038 | Zbl 1086.11015

[2] G. Badkobeh and M. Crochemore, Finite-Repetition threshold for infinite ternary words, in 8th International Conference WORDS 2011, edited by P. Ambroz, S. Holub and Z. Masáková. EPTCS 63 (2011) 37-43.

[3] J. Cassaigne, Unavoidable binary patterns. Acta Informatica 30 (1993) 385-395. | MR 1227889 | Zbl 0790.68096

[4] C. Choffrut and J. Karhumäki, Combinatorics of words, in vol. 1 of Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa. Springer (1997) 329-438. | MR 1469998 | Zbl 0866.68057

[5] K. Culik, J. Karhumäki and A. Lepistö, Alternating iteration of morphisms and the Kolakoski sequence, in Lindenmayer Systems, edited by G. Rozenberg and A. Salomaa. Springer (1992) 93-106. | MR 1226686 | Zbl 0766.68073

[6] J.D. Currie, Open problems in pattern avoidance. Amer. Math. Monthly 100 (1993) 790-793. | MR 1542406 | Zbl 0798.68139

[7] F.M. Dekking, Strongly non-repetitive sequences and progression-free sets. J. Combin. Theory Ser. A 27 (1979) 181-185. | MR 542527 | Zbl 0437.05011

[8] A.A. Evdokimov, Strongly asymmetric sequences generated by a finite number of symbols. Dokl. Akad. Nauk SSSR 179 (1968) 1268-1271; English translation in Soviet Math. Dokl. 9 (1968) 536-539. | MR 234842 | Zbl 0186.01504

[9] M. Huova and J. Karhumäki, On Unavoidability of k-abelian Squares in Pure Morphic Words. J. Integer Sequences, being bublished. | Zbl 1285.68135

[10] M. Huova, J. Karhumäki and A. Saarela, Problems in between words and abelian words: k-abelian avoidability, in Formal and Natural Computing Honoring the 80th Birthday of Andrzej Ehrenfeucht, edited by G. Rozenberg and A. Salomaa. Theoret. Comput. Sci. 454 (2012) 172-177. | MR 2966632 | Zbl 1280.68149

[11] M. Huova, J. Karhumäki, A. Saarela and K. Saari, Local squares, periodicity and finite automata, in Rainbow of Computer Science, edited by C.S. Calude, G. Rozenberg and A. Salomaa. Vol. 6570 of Lect. Notes Comput. Sci. Springer (2011) 90-101. | MR 2805552 | Zbl pre05900368

[12] V. Keränen, Abelian squares are avoidable on 4 letters, in Proc. ICALP 1992, edited by W. Kuich. Vol. 623 of Lect. Notes Comput. Sci. Springer (1992) 41-52. | MR 1250629

[13] M. Lothaire, Combinatorics on words, Addison-Wesley (1983). | MR 675953 | Zbl 0514.20045

[14] M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002). | MR 1905123 | Zbl 1221.68183

[15] R. Mercas and A. Saarela, 5-abelian cubes are avoidable on binary alphabets, in the conference the 14th Mons Days of Theoretical Computer Science (2012). | Zbl pre06182460

[16] P.A.B. Pleasant, Non-repetitive sequences. Proc. Cambridge Philos. Soc. 68 (1970) 267-274. | MR 265173 | Zbl 0237.05010

[17] A. Thue, Über unendliche Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 7 (1906) 1-22. | JFM 39.0283.01

[18] A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1-67. | JFM 44.0462.01