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Self-shuffling Words

  • Émilie Charlier
  • Teturo Kamae
  • Svetlana Puzynina
  • Luca Q. Zamboni
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)

Abstract

In this paper we introduce and study a new property of infinite words which is invariant under the action of a morphism: We say an infinite word \(x\in \mathbb{A}^{\mathbb N},\) defined over a finite alphabet \(\mathbb{A}\), is self-shuffling if x admits factorizations: \(x=\prod_{i=1}^\infty U_iV_i=\prod_{i=1}^\infty U_i=\prod_{i=1}^\infty V_i\) with \(U_i,V_i \in \mathbb{A}^+.\) In other words, there exists a shuffle of x with itself which reproduces x. The morphic image of any self-shuffling word is again self-shuffling. We prove that many important and well studied words are self-shuffling: This includes the Thue-Morse word and all Sturmian words (except those of the form aC where a ∈ {0,1} and C is a characteristic Sturmian word). We further establish a number of necessary conditions for a word to be self-shuffling, and show that certain other important words (including the paper-folding word and infinite Lyndon words) are not self-shuffling. In addition to its morphic invariance, which can be used to show that one word is not the morphic image of another, this new notion has other unexpected applications: For instance, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, which characterizes pure morphic Sturmian words in the orbit of the characteristic.

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References

  1. 1.
    Allouche, J.-P., Shallit, J.: The ubiquitous Prouhet-Thue-Morse sequence. In: Ding, C., Helleseth, T., Niederreiter, H. (eds.) Proceedings of Sequences and Their Applications, SETA 1998, pp. 1–16. Springer (1999)Google Scholar
  2. 2.
    Allouche, J.-P., Shallit, J.: Automatic sequences. In: Theory, Applications, Generalizations. Cambridge University Press (2003)Google Scholar
  3. 3.
    Berthé, V., Ei, H., Ito, S., Rao, H.: On substitution invariant Sturmian words: an application of Rauzy fractals. Theor. Inform. Appl. 41, 329–349 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cassaigne, J., Karhumäki, J.: Toeplitz Words, Generalized Periodicity and Periodically Iterated Morphisms. European J. Combin. 18, 497–510 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Henshall, D., Rampersad, N., Shallit, J.: Shuffling and unshuffling. Bull. EATCS 107, 131–142 (2012)Google Scholar
  6. 6.
    Fagnot, I.: A little more about morphic Sturmian words. Theor. Inform. Appl. 40, 511–518 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Lothaire, M.: Algebraic Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, U.K (2002)zbMATHCrossRefGoogle Scholar
  8. 8.
    Morse, M., Hedlund, G.A.: Symbolic dynamics II: Sturmian sequences. Amer. J. Math. 62, 1–42 (1940)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Thue, A.: Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I Math-Nat. Kl. 7, 1–22 (1906)Google Scholar
  10. 10.
    Yasutomi, S.-I.: On sturmian sequences which are invariant under some substitutions. In: Kanemitsu, S., et al. (eds.) Number Theory and Its Applications, Proceedings of the Conference held at the RIMS, Kyoto, Japan, November 10-14, 1997, pp. 347–373. Kluwer Acad. Publ., Dordrecht (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Émilie Charlier
    • 1
  • Teturo Kamae
    • 2
  • Svetlana Puzynina
    • 3
    • 5
  • Luca Q. Zamboni
    • 4
    • 5
  1. 1.Département de MathématiqueUniversité de LiègeBelgium
  2. 2.Advanced Mathematical InstituteOsaka City UniversityJapan
  3. 3.Sobolev Institute of MathematicsNovosibirskRussia
  4. 4.Institut Camille JordanUniversité Lyon 1France
  5. 5.FUNDIMUniversity of TurkuFinland

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