On Words with the Zero Palindromic Defect

  • Edita Pelantová
  • Štěpán StarostaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)


We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word.


Palindrome Palindromic defect Rich words 



The authors acknowledge financial support by the Czech Science Foundation grant GAČR 13-03538S.


  1. 1.
    Bašić, B.: On highly potential words. Eur. J. Combin. 34(6), 1028–1039 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baláži, P., Masáková, Z., Pelantová, E.: Factor versus palindromic complexity of uniformly recurrent infinite words. Theoret. Comput. Sci. 380(3), 266–275 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balková, L., Pelantová, E., Starosta, Š.: Infinite words with finite defect. Adv. Appl. Math. 47(3), 562–574 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balková, L., Pelantová, E., Starosta, Š.: Proof of the Brlek-Reutenauer conjecture. Theoret. Comput. Sci. 475, 120–125 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Massé, A.B., Brlek, S., Frosini, A., Labbé, S., Rinaldi, S.: Reconstructing words from a fixed palindromic length sequence. In: Ausiello, G., Karhumäki, J., Mauri, G., Ong, L. (eds.) TCS 2008. IIFIP, vol. 273, pp. 101–114. Springer, Boston, MA (2008). doi: 10.1007/978-0-387-09680-3_7 CrossRefGoogle Scholar
  6. 6.
    Blondin Massé, A., Brlek, S., Garon, A., Labbé, S.: Combinatorial properties of \(f\)-palindromes in the Thue-Morse sequence. Pure Math. Appl. 19(2–3), 39–52 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Blondin Massé, A., Brlek, S., Labbé, S., Vuillon, L.: Palindromic complexity of codings of rotations. Theoret. Comput. Sci. 412(46), 6455–6463 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brlek, S., Reutenauer, C.: Complexity and palindromic defect of infinite words. Theoret. Comput. Sci. 412(4–5), 493–497 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brlek, S., Hamel, S., Nivat, M., Reutenauer, C.: On the palindromic complexity of infinite words. Int. J. Found. Comput. Sci. 15(2), 293–306 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bucci, M., Vaslet, E.: Palindromic defect of pure morphic aperiodic words. In: Proceedings of the 14th Mons Days of Theoretical Computer Science (2012)Google Scholar
  11. 11.
    Bucci, M., De Luca, A., Glen, A., Zamboni, L.Q.: A connection between palindromic and factor complexity using return words. Adv. Appl. Math. 42(1), 60–74 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bucci, M., De Luca, A., Glen, A., Zamboni, L.Q.: A new characteristic property of rich words. Theoret. Comput. Sci. 410(30–32), 2860–2863 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bucci, M., Luca, A., Luca, A.: Rich and periodic-like words. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 145–155. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02737-6_11 CrossRefGoogle Scholar
  14. 14.
    Droubay, X., Justin, J., Pirillo, G.: Episturmian words and some constructions of de Luca and Rauzy. Theoret. Comput. Sci. 255(1–2), 539–553 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Droubay, X., Pirillo, G.: Palindromes and Sturmian words. Theoret. Comput. Sci. 223(1–2), 73–85 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Glen, A., Justin, J., Widmer, S., Zamboni, L.Q.: Palindromic richness. Eur. J. Combin. 30(2), 510–531 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guo, C., Shallit, J., Shur, A.M.: Palindromic rich words and run-length encodings. Inf. Process. Lett. 116(12), 735–738 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Harju, T., Vesti, J., Zamboni, L.Q.: On a question of Hof, Knill and Simon on palindromic substitutive systems. Monatsh. Math. 179(3), 379–388 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hof, A., Knill, O., Simon, B.: Singular continuous spectrum for palindromic Schrödinger operators. Commun. Math. Phys. 174, 149–159 (1995)CrossRefzbMATHGoogle Scholar
  20. 20.
    Jajcayová, T., Pelantová, E., Starosta, Š.: Palindromic closures using multiple antimorphisms. Theoret. Comput. Sci. 533, 37–45 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Labbé, S., Pelantová, E.: Palindromic sequences generated from marked morphisms. Eur. J. Combin. 51, 200–214 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Labbé, S., Pelantová, E., Starosta, Š.: On the zero defect conjecture. Eur. J. Comb. 62, 132–146 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Labbé, S.: A counterexample to a question of Hof, Knill and Simon. Electron. J. Combin. 21 (2014). Paper #P3.11Google Scholar
  24. 24.
    de Luca, A., Glen, A., Zamboni, L.Q.: Rich, sturmian, and trapezoidal words. Theoret. Comput. Sci. 407(1), 569–573 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Masáková, Z., Pelantová, E., Starosta, Š.: Exchange of three intervals: substitutions and palindromicity. Eur. J. Combin. 62, 217–231 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pelantová, E., Starosta, Š.: Languages invariant under more symmetries: overlapping factors versus palindromic richness. Discret. Math. 313, 2432–2445 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pelantová, E., Starosta, Š.: Palindromic richness for languages invariant under more symmetries. Theor. Comput. Sci. 518, 42–63 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pelantová, E., Starosta, Š.: Constructions of words rich in palindromes and pseudopalindromes. Discret. Math. Theoret. Comput. Sci. 18(3), 1–26 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Restivo, A., Rosone, G.: Balancing and clustering of words in the Burrows-Wheeler transform. Theoret. Comput. Sci. 412(27), 3019–3032 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rukavicka, J.: On number of rich words (2017). Preprint available at arXiv:1701.07778
  31. 31.
    Starosta, Š.: Generalized Thue-Morse words and palindromic richness. Kybernetika 48(3), 361–370 (2012)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Starosta, Š.: Morphic images of episturmian words having finite palindromic defect. Eur. J. Combin. 51, 359–371 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tan, B.: Mirror substitutions and palindromic sequences. Theoret. Comput. Sci. 389(1–2), 118–124 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Vesti, J.: Extensions of rich words. Theoret. Comput. Sci. 548, 14–24 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePragueCzech Republic
  2. 2.Department of Applied Mathematics, Faculty of Information TechnologyCzech Technical University in PraguePragueCzech Republic

Personalised recommendations