The Lengths for Which Bicrucial Square-Free Permutations Exist

Abstract

: A square is a factor S = (S1; S2) where S1 and S2 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.