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On Growth and Fluctuation of k-Abelian Complexity

  • Julien Cassaigne
  • Juhani Karhumäki
  • Aleksi Saarela
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9139)

Abstract

An extension of abelian complexity, so called k-abelian complexity, has been considered recently in a number of articles. This paper considers two particular aspects of this extension: First, how much the complexity can increase when moving from a level k to the next one. Second, how much the complexity of a given word can fluctuate. For both questions we give optimal solutions.

References

  1. 1.
    Balogh, J., Bollobás, B.: Hereditary properties of words. RAIRO Inform. Theor. Appl. 39(1), 49–65 (2005)MATHCrossRefGoogle Scholar
  2. 2.
    Cassaigne, J., Karhumäki, J.: Toeplitz words, generalized periodicity and periodically iterated morphisms. Eur. J. Comb. 18(5), 497–510 (1997)MATHCrossRefGoogle Scholar
  3. 3.
    Coven, E.M., Hedlund, G.A.: Sequences with minimal block growth. Math. Syst. Theory 7, 138–153 (1973)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Dekking, M.: Strongly nonrepetitive sequences and progression-free sets. J. Combin. Theory Ser. A 27(2), 181–185 (1979)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Greinecker, F.: On the 2-abelian complexity of Thue-Morse subwords (Preprint). arXiv:1404.3906
  6. 6.
    Harmaala, E.: Sanojen ekvivalenssiluokkien laskentaa (2010) (manuscript)Google Scholar
  7. 7.
    Huova, M., Karhumäki, J., Saarela, A.: Problems in between words and abelian words: \(k\)-abelian avoidability. Theoret. Comput. Sci. 454, 172–177 (2012)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Huova, M., Karhumäki, J., Saarela, A., Saari, K.: Local squares, periodicity and finite automata. In: Calude, C.S., Rozenberg, G., Salomaa, A. (eds.) Rainbow of Computer Science. LNCS, vol. 6570, pp. 90–101. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  9. 9.
    Jacobs, K., Keane, M.: \(0-1\)-sequences of Toeplitz type. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13(2), 123–131 (1969)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Karhumäki, J., Saarela, A., Zamboni, L.Q.: On a generalization of abelian equivalence and complexity of infinite words. J. Combin. Theory Ser. A 120(8), 2189–2206 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Karhumäki, J., Saarela, A., Zamboni, L.Q.: Variations of the Morse-Hedlund theorem for k-abelian equivalence. In: Shur, A.M., Volkov, M.V. (eds.) DLT 2014. LNCS, vol. 8633, pp. 203–214. Springer, Heidelberg (2014) Google Scholar
  12. 12.
    Keränen, V.: Abelian squares are avoidable on 4 letters. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 41–152. Springer, Heidelberg (1992) CrossRefGoogle Scholar
  13. 13.
    Mercaş, R., Saarela, A.: 3-abelian cubes are avoidable on binary alphabets. In: Béal, M.-P., Carton, O. (eds.) DLT 2013. LNCS, vol. 7907, pp. 374–383. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  14. 14.
    Mercaş, R., Saarela, A.: \(5\)-abelian cubes are avoidable on binary alphabets. RAIRO Inform. Theor. Appl. 48(4), 467–478 (2014)MATHCrossRefGoogle Scholar
  15. 15.
    Morse, M., Hedlund, G.A.: Symbolic dynamics. Amer. J. Math. 60(4), 815–866 (1938)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Morse, M., Hedlund, G.A.: Symbolic dynamics II: Sturmian trajectories. Amer. J. Math. 62(1), 1–42 (1940)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Parreau, A., Rigo, M., Rowland, E., Vandomme, E.: A new approach to the 2-regularity of the \(l\)-abelian complexity of 2-automatic sequences. Electron. J. Combin. 22(1), P1.27 (2015)MathSciNetGoogle Scholar
  18. 18.
    Rao, M.: On some generalizations of abelian power avoidability (Manuscript)Google Scholar
  19. 19.
    Thue, A.: Über unendliche zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. 7, 1–22 (1906)Google Scholar
  20. 20.
    Thue, A.: Über die gegenseitige lage gleicher teile gewisser zeichen-reihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. 1, 1–67 (1912)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Julien Cassaigne
    • 1
  • Juhani Karhumäki
    • 2
  • Aleksi Saarela
    • 2
  1. 1.Institut de Mathmatiques de LuminyMarseille Cedex 9France
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

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