On square-free permutations

Sergey Avgustinovich, Sergey Kitaev, Artem Pyatkin, Alexander Valyuzhenich

Research output: Contribution to journalArticle


A permutation is square-free if it does not contain two consecutive factors of length more than one that coincide in the reduced form (as patterns). We prove that the number of square-free permutations of length n is nn(1􀀀"n) where "n ! 0 when n ! 1. A permutation of length n is crucial with respect to squares if it avoids squares but any extension of it to the right, to a permutation of length n+1, contains a square. A permutation is maximal with respect to squares if both the permutation and its reverse are crucial with respect
to squares. We prove that there exist crucial permutations with respect to squares of any length at least 7, and there exist maximal permutations with respect to squares of odd lengths 8k+1; 8k+5; 8k+7 for k 1.
Original languageEnglish
Pages (from-to)3-10
Number of pages8
JournalJournal of Automata, Languages and Combinatorics
Issue number1
Publication statusPublished - 2011



  • square freeness
  • consecutive pattern
  • enumeration
  • crucial word
  • maximal word
  • permutation

Cite this

Avgustinovich, S., Kitaev, S., Pyatkin, A., & Valyuzhenich, A. (2011). On square-free permutations. Journal of Automata, Languages and Combinatorics, 16(1), 3-10.