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On Growth and Fluctuation of k-Abelian Complexity

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Computer Science -- Theory and Applications (CSR 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9139))

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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.

Supported by the Academy of Finland under grant 257857.

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References

  1. Balogh, J., Bollobás, B.: Hereditary properties of words. RAIRO Inform. Theor. Appl. 39(1), 49–65 (2005)

    Article  MATH  Google Scholar 

  2. Cassaigne, J., Karhumäki, J.: Toeplitz words, generalized periodicity and periodically iterated morphisms. Eur. J. Comb. 18(5), 497–510 (1997)

    Article  MATH  Google Scholar 

  3. Coven, E.M., Hedlund, G.A.: Sequences with minimal block growth. Math. Syst. Theory 7, 138–153 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dekking, M.: Strongly nonrepetitive sequences and progression-free sets. J. Combin. Theory Ser. A 27(2), 181–185 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  5. Greinecker, F.: On the 2-abelian complexity of Thue-Morse subwords (Preprint). arXiv:1404.3906

  6. Harmaala, E.: Sanojen ekvivalenssiluokkien laskentaa (2010) (manuscript)

    Google Scholar 

  7. Huova, M., Karhumäki, J., Saarela, A.: Problems in between words and abelian words: \(k\)-abelian avoidability. Theoret. Comput. Sci. 454, 172–177 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  9. Jacobs, K., Keane, M.: \(0-1\)-sequences of Toeplitz type. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13(2), 123–131 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Keränen, V.: Abelian squares are avoidable on 4 letters. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 41–152. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  14. Mercaş, R., Saarela, A.: \(5\)-abelian cubes are avoidable on binary alphabets. RAIRO Inform. Theor. Appl. 48(4), 467–478 (2014)

    Article  MATH  Google Scholar 

  15. Morse, M., Hedlund, G.A.: Symbolic dynamics. Amer. J. Math. 60(4), 815–866 (1938)

    Article  MathSciNet  Google Scholar 

  16. Morse, M., Hedlund, G.A.: Symbolic dynamics II: Sturmian trajectories. Amer. J. Math. 62(1), 1–42 (1940)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  18. Rao, M.: On some generalizations of abelian power avoidability (Manuscript)

    Google Scholar 

  19. Thue, A.: Über unendliche zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. 7, 1–22 (1906)

    Google Scholar 

  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 

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Author information

Authors and Affiliations

  1. Institut de Mathmatiques de Luminy, Case 907, 13288, Marseille Cedex 9, France

    Julien Cassaigne

  2. Department of Mathematics and Statistics, University of Turku, 20014, Turku, Finland

    Juhani Karhumäki & Aleksi Saarela

Authors
  1. Julien Cassaigne
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  2. Juhani Karhumäki
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  3. Aleksi Saarela
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Corresponding author

Correspondence to Aleksi Saarela .

Editor information

Editors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Lomonosov Moscow State University and National Research University Higher School of Economics, Moscow, Russia

    Lev D. Beklemishev

  2. Moscow Institute of Physics and Technology, Moscow, Russia and Kazan (Volga Region) Federal University, Kazan, Kabardino-Balkar.Rep, Russia

    Daniil V. Musatov

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Cassaigne, J., Karhumäki, J., Saarela, A. (2015). On Growth and Fluctuation of k-Abelian Complexity. In: Beklemishev, L., Musatov, D. (eds) Computer Science -- Theory and Applications. CSR 2015. Lecture Notes in Computer Science(), vol 9139. Springer, Cham. https://doi.org/10.1007/978-3-319-20297-6_8

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  • DOI: https://doi.org/10.1007/978-3-319-20297-6_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20296-9

  • Online ISBN: 978-3-319-20297-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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