Abelian Pattern Avoidance in Partial Words

  • Francine Blanchet-Sadri
  • Sean Simmons
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


Pattern avoidance is an important topic in combinatorics on words which dates back to the beginning of the twentieth century when Thue constructed an infinite word over a ternary alphabet that avoids squares, i.e., a word with no two adjacent identical factors. This result finds applications in various algebraic contexts where more general patterns than squares are considered. On the other hand, Erdős raised the question as to whether there exists an infinite word that avoids abelian squares, i.e., a word with no two adjacent factors being permutations of one another. Although this question was answered affirmately years later, knowledge of abelian pattern avoidance is rather limited. Recently, (abelian) pattern avoidance was initiated in the more general framework of partial words, which allow for undefined positions called holes. Here, we investigate conditions for a pattern to be abelian avoidable by a partial word with finitely or infinitely many holes.


Distinct Variable Alphabet Size Pigeonhole Principle Partial Word Binary 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berstel, J., Boasson, L.: Partial words and a theorem of Fine and Wilf. Theoretical Computer Science 218, 135–141 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Blanchet-Sadri, F.: Algorithmic Combinatorics on Partial Words. Chapman & Hall/CRC Press, Boca Raton, FL (2008)zbMATHGoogle Scholar
  3. 3.
    Blanchet-Sadri, F., Kim, J.I., Mercaş, R., Severa, W., Simmons, S., Xu, D.: Avoiding abelian squares in partial words. Journal of Combinatorial Theory, Series A 119, 257–270 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Blanchet-Sadri, F., Simmons, S., Xu, D.: Abelian repetitions in partial words. Advances in Applied Mathematics 48, 194–214 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Currie, J., Linek, V.: Avoiding patterns in the abelian sense. Canadian Journal of Mathematics 53, 696–714 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Currie, J., Visentin, T.: On abelian 2-avoidable binary patterns. Acta Informatica 43, 521–533 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Currie, J., Visentin, T.: Long binary patterns are abelian 2-avoidable. Theoretical Computer Science 409, 432–437 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Currie, J.D.: Pattern avoidance: themes and variations. Theoretical Computer Science 339, 7–18 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dekking, F.M.: Strongly non-repetitive sequences and progression-free sets. Journal of Combinatorial Theory, Series A 27(2), 181–185 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Erdős, P.: Some unsolved problems. Magyar Tudományos Akadémia Matematikai Kutató Intézete Közl. 6, 221–254 (1961)Google Scholar
  11. 11.
    Keränen, V.: Abelian Squares are Avoidable on 4 Letters. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 41–52. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  12. 12.
    Leupold, P.: Partial Words for DNA Coding. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 224–234. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  14. 14.
    Thue, A.: Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiana 7, 1–22 (1906)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Francine Blanchet-Sadri
    • 1
  • Sean Simmons
    • 2
  1. 1.Department of Computer ScienceUniversity of North CarolinaGreensboroUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations