Skip to main content

Overpals, Underlaps, and Underpals

  • Conference paper
  • First Online:
  • 510 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10432))

Abstract

An overlap in a word is a factor of the form axaxa, where x is a (possibly empty) word and a is a single letter; these have been well-studied since Thue’s landmark paper of 1906. In this note we consider three new variations on this well-known definition and some consequences.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Allouche, J.P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  2. Berstel, J.: Axel Thue’s papers on repetitions in words: a translation, No. 20. In: Publications du Laboratoire de Combinatoire et d’Informatique Mathématique, Université du Québec à Montréal, February 1995

    Google Scholar 

  3. Blondel, V.D., Cassaigne, J., Jungers, R.M.: On the number of \(\alpha \)-power-free binary words for \(2 < \alpha \le 7/3\). Theoret. Comput. Sci. 410, 2823–2833 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cassaigne, J.: Counting overlap-free binary words. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds.) STACS 1993. LNCS, vol. 665, pp. 216–225. Springer, Heidelberg (1993). doi:10.1007/3-540-56503-5_24

    Chapter  Google Scholar 

  5. Dejean, F.: Sur un théorème de Thue. J. Combin. Theor. Ser. A 13, 90–99 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Goulden, I., Jackson, D.: An inversion theorem for cluster decompositions of sequences with distinguished subsequences. J. London Math. Soc. 20, 567–576 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jungers, R.M., Protasov, V.Y., Blondel, V.D.: Overlap-free words and spectra of matrices. Theor. Comput. Sci. 410, 3670–3684 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lothaire, M.: Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  9. Mousavi, H.: Automatic theorem proving in Walnut (2016). https://arxiv.org/abs/1603.06017

  10. Noonan, J., Zeilberger, D.: DAVID_IAN Maple package (1999). http://www.math.rutgers.edu/~zeilberg/gj.html

  11. Noonan, J., Zeilberger, D.: The Goulden-Jackson cluster method: extensions, applications, and implementations. J. Differ. Equ. Appl. 5, 355–377 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rampersad, N.: Overlap-free words and generalizations. Ph.D. thesis, University of Waterloo (2008)

    Google Scholar 

  13. Sollami, M., Douglas, C.C., Liebmann, M.: An improved lower bound on the number of ternary squarefree words. J. Integer Sequences 19, 3 (2016). Article 16.6.7

    MathSciNet  MATH  Google Scholar 

  14. Tan, B., Wen, Z.Y.: Invertible substitutions and Sturmian sequences. Eur. J. Comb. 24, 983–1002 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Thue, A.: Über unendliche Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 7, 1–22 (1906). Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, 1977, pp. 139–158

    MATH  Google Scholar 

  16. Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1, 1–67 (1912). Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, 1977, pp. 413–478

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jeffrey Shallit .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Rajasekaran, A., Rampersad, N., Shallit, J. (2017). Overpals, Underlaps, and Underpals. In: Brlek, S., Dolce, F., Reutenauer, C., Vandomme, É. (eds) Combinatorics on Words. WORDS 2017. Lecture Notes in Computer Science(), vol 10432. Springer, Cham. https://doi.org/10.1007/978-3-319-66396-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66396-8_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66395-1

  • Online ISBN: 978-3-319-66396-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics