The lengths for which bicrucial square-free permutations exist

Abstract

A square is a factor S = (S 1 ; S 2 ) where S 1 and S 2 have the same pattern, and a permutation is said to be square-free if it contains no non-trivial squares. The permutation is further said to be bicrucial if every extension to the left or right contains a square. We completely classify for which n there exists a bicrucial square-free permutation of length n.