On ternary Dejean words avoiding 010

Pascal Ochem
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 3, Pages 545–548
DOI: 10.7546/nntdm.2023.29.3.545-548
Full paper (PDF, 187 Kb)


Authors and affiliations

Pascal Ochem
LIRMM, Université de Montpellier, CNRS
Montpellier, France


Thue has shown the existence of three types of infinite square-free words over \left\{\texttt{0},\texttt{1},\texttt{2}\right\} avoiding the factor \texttt{010}. They respectively avoid \left\{\texttt{010}, \texttt{212}\right\}, \left\{\texttt{010},\texttt{101}\right\}, and \left\{\texttt{010},\texttt{020}\right\}. Also Dejean constructed an infinite \left(\tfrac74^+\right)-free ternary word. A word is d-directed if it does not contain both a factor of length d and its mirror image. We show that there exist exponentially many \left(\tfrac74^+\right)-free 180-directed ternary words avoiding \texttt{010}. Moreover, there does not exist an infinite \left(\tfrac74^+\right)-free 179-directed ternary word avoiding \texttt{010}.


  • Word
  • Repetitions

2020 Mathematics Subject Classification

  • 68R15


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Manuscript history

  • Received: 3 March 2023
  • Revised: 17 July 2023
  • Accepted: 25 July 2023
  • Online First: 27 July 2023

Copyright information

Ⓒ 2023 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Ochem, P. (2023). On ternary Dejean words avoiding 010. Notes on Number Theory and Discrete Mathematics, 29(3), 545-548, DOI: 10.7546/nntdm.2023.29.3.545-548.

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