A Stronger Square Conjecture on Binary Words

  • Nataša Jonoska
  • Florin Manea
  • Shinnosuke Seki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8327)


We propose a stronger conjecture regarding the number of distinct squares in a binary word. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a word is upper bounded by the length of the word. Here, we conjecture that in the case of a word of length n over the alphabet {a,b}, the number of distinct squares is upper bounded by \(\frac{2k-1}{2k+2}n\), where k is the least of the number of a’s and the number of b’s. We support the conjecture by showing its validity for several classes of binary words. We also prove that the bound is tight.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fraenkel, A.S., Simpson, J.: How many squares can a string contain? Journal of Combinatorial Theory, Series A 82, 112–120 (1998)Google Scholar
  2. 2.
    Deza, A., Franek, F., Jiang, M.: A d-step approach for distinct squares in strings. In: Giancarlo, R., Manzini, G. (eds.) CPM 2011. LNCS, vol. 6661, pp. 77–89. Springer, Heidelberg (2011)Google Scholar
  3. 3.
    Ilie, L.: A note on the number of squares in a word. Theoretical Computer Science 380, 373–376 (2007)Google Scholar
  4. 4.
    Fan, K., Puglisi, S.J., Smyth, W.F., Turpin, A.: A new periodicity lemma. SIAM Journal of Discrete Mathematics 20(3), 656–668 (2005)Google Scholar
  5. 5.
    Ilie, L.: A simple proof that a word of length n has at most 2n distinct squares. Journal of Combinatorial Theory, Series A 112(1), 163–164 (2005)Google Scholar
  6. 6.
    Kopylova, E., Smyth, W.F.: The three squares lemma revisited. Journal of Discrete Algorithms 11, 3–14 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Nataša Jonoska
    • 1
  • Florin Manea
    • 2
  • Shinnosuke Seki
    • 3
    • 4
  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  2. 2.Institut für InformatikChristian-Albrechts-Universität zu KielKielGermany
  3. 3.Helsinki Institute for Information Technology (HIIT)Finland
  4. 4.Department of Information and Computer ScienceAalto UniversityAaltoFinland

Personalised recommendations