Palindromic Length in Free Monoids and Free Groups

  • Aleksi SaarelaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)


Palindromic length of a word is defined as the smallest number n such that the word can be written as a product of n palindromes. It has been conjectured that every aperiodic infinite word has factors of arbitrarily high palindromic length. A stronger variant of this conjecture claims that every aperiodic infinite word has also prefixes of arbitrarily high palindromic length. We prove that these two conjectures are equivalent. More specifically, we prove that if every prefix of a word is a product of n palindromes, then every factor of the word is a product of 2n palindromes. Our proof quite naturally leads us to compare the properties of palindromic length in free monoids and in free groups. For example, the palindromic lengths of a word and its conjugate can be arbitrarily far apart in a free monoid, but in a free group they are almost the same.


Combinatorics on words Palindrome Free group 


  1. 1.
    Allouche, J.P., Baake, M., Cassaigne, J., Damanik, D.: Palindrome complexity. Theoret. Comput. Sci. 291(1), 9–31 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bardakov, V.G., Gongopadhyay, K.: Palindromic width of finitely generated solvable groups. Comm. Algebra 43(11), 4809–4824 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bardakov, V.G., Shpilrain, V., Tolstykh, V.: On the palindromic and primitive widths of a free group. J. Algebra 285(2), 574–585 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blondin Massé, A., Brlek, S., Labbé, S.: Palindromic lacunas of the Thue-Morse word. In: Proceedings of GASCom, pp. 53–67 (2008)Google Scholar
  5. 5.
    Borchert, A., Rampersad, N.: Words with many palindrome pair factors. Electron. J. Comb. 22(4), P4.23 (2015)Google Scholar
  6. 6.
    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
  7. 7.
    Bucci, M., Richomme, G.: Greedy palindromic lengths (Preprint).
  8. 8.
    Fici, G., Gagie, T., Kärkkäinen, J., Kempa, D.: A subquadratic algorithm for minimum palindromic factorization. J. Discrete Algorithms 28, 41–48 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fink, E.: Palindromic width of wreath products. J. Algebra 471, 1–12 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frid, A.E., Puzynina, S., Zamboni, L.Q.: On palindromic factorization of words. Adv. Appl. Math. 50(5), 737–748 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Glen, A., Justin, J., Widmer, S., Zamboni, L.Q.: Palindromic richness. Eur. J. Comb. 30(2), 510–531 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guo, C., Shallit, J., Shur, A.M.: On the combinatorics of palindromes and antipalindromes (Preprint).
  13. 13.
    Holub, Š., Müller, M.: Fully bordered words. Theoret. Comput. Sci. 684, 53–58 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    I, T., Sugimoto, S., Inenaga, S., Bannai, H., Takeda, M.: Computing palindromic factorizations and palindromic covers on-line. In: Kulikov, A.S., Kuznetsov, S.O., Pevzner, P. (eds.) CPM 2014. LNCS, vol. 8486, pp. 150–161. Springer, Cham (2014). doi: 10.1007/978-3-319-07566-2_16 Google Scholar
  15. 15.
    Ravsky, O.: On the palindromic decomposition of binary words. J. Autom. Lang. Comb. 8(1), 75–83 (2003)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

Personalised recommendations