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)


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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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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