Abelian Primitive Words

  • Michael Domaratzki
  • Narad Rampersad
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)


We investigate Abelian primitive words, which are words that are not Abelian powers. We show the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive in linear time. Also different from classical primitive words, we find that a word may have more than one Abelian root. We also consider enumeration of Abelian primitive words.


Formal Language Linear Time Algorithm Primitive Root Formal Language Theory Lyndon Word 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Anderson, I.: On primitive sequences. Journal London Math. Soc. 42, 137–148 (1967)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anderson, I.: Combinatorics of finite sets. Dover Publications, Mineola (2002)MATHGoogle Scholar
  3. 3.
    Bach, E., Shallit, J.: Algorithmic Number Theory, vol. 1. MIT Press, Cambridge (1997)MATHGoogle Scholar
  4. 4.
    Berstel, J., Boasson, L.: The set of Lyndon words is not context-free. Bull. EATCS 63, 139–140 (1997)MathSciNetMATHGoogle Scholar
  5. 5.
    Constantinescu, S., Ilie, L.: Fine and Wilf’s theorem for Abelian periods. Bull. EATCS 89, 167–170 (2006)MathSciNetMATHGoogle Scholar
  6. 6.
    Czeizler, E., Kari, L., Seki, S.: On a special class of primitive words. Theoretical Computer Science 411, 617–630 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    de Bruijn, N., van Ebbenhorst Tengbergen, C., Kruyswijk, D.: On the set of divisors of a number. Nieuw Arch. Wiskunde 23, 191–193 (1951)MathSciNetMATHGoogle Scholar
  8. 8.
    Dömösi, P., Horváth, S., Ito, M., Kászonyi, L., Katsura, M.: Formal languages consisting of primitive words. In: Ésik, Z. (ed.) FCT 1993. LNCS, vol. 710, pp. 194–203. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  9. 9.
    Gabarró, J.: Some applications of the interchange lemma. Bull. EATCS 25, 19–21 (1985)Google Scholar
  10. 10.
    Hardy, G., Wright, E.: An Introduction to the Theory of Numbers, 5th edn. Oxford Science Publications, Oxford (2000)Google Scholar
  11. 11.
    Lothaire, M.: Combinatorics on words. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  12. 12.
    Petersen, H.: The ambiguity of primitive words. In: Enjalbert, P., Mayr, E., Wagner, K. (eds.) STACS 1994. LNCS, vol. 775, pp. 679–690. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  13. 13.
    Richmond, L.B., Shallit, J.: Counting Abelian squares. Elec. J. Combinatorics 16, R72 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Rozenberg, G., Salomaa, A.: Handbook of Formal Languages, vol. 1. Springer, Heidelberg (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michael Domaratzki
    • 1
  • Narad Rampersad
    • 2
  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada
  2. 2.Department of MathematicsUniversity of LiègeLiègeBelgium

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