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Existence of an infinite ternary 64-abelian square-freeword

Published online by Cambridge University Press:  10 July 2014

Mari Huova
Mari Huova*
Affiliation:
Department of Mathematics and Statistics & TUCS, University of Turku, 20014 Turku, Finland.. mari.huova@utu.fi
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Abstract

We consider a recently defined notion of k-abelian equivalence of words byconcentrating on avoidance problems. The equivalence class of a word depends on thenumbers of occurrences of different factors of length k for a fixed naturalnumber k andthe prefix of the word. We have shown earlier that over a ternary alphabet k-abelian squares cannot beavoided in pure morphic words for any natural number k. Nevertheless,computational experiments support the conjecture that even 3-abelian squares can beavoided over ternary alphabets. In this paper we establish the first avoidance resultshowing that by choosing k to be large enough we have an infinitek-abeliansquare-free word over three letter alphabet. In addition, this word can be obtained as amorphic image of a pure morphic word.

Keywords

Combinatorics on wordsk-abelianequivalencesquare-freeness

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References

J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003).
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.
Cassaigne, J., Unavoidable binary patterns. Acta Informatica 30 (1993) 385–395. Google Scholar
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.
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.
Currie, J.D., Open problems in pattern avoidance. Amer. Math. Monthly 100 (1993) 790–793. Google Scholar
Dekking, F.M., Strongly non-repetitive sequences and progression-free sets. J. Combin. Theory Ser. A 27 (1979) 181–185. Google Scholar
Evdokimov, A.A., 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. Google Scholar
M. Huova and J. Karhumäki, On Unavoidability of k-abelian Squares in Pure Morphic Words. J. Integer Sequences, being bublished.
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.
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.
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.
M. Lothaire, Combinatorics on words, Addison-Wesley (1983).
M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002).
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).
Pleasant, P.A.B., Non-repetitive sequences. Proc. Cambridge Philos. Soc. 68 (1970) 267–274. Google Scholar
Thue, A., Über unendliche Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 7 (1906) 1–22. Google Scholar
Thue, A., Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1–67. Google Scholar