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S-Crucial and Bicrucial Permutations with Respect to Squares

2015
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Journal of Integer Sequences
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unpublished

A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A permutation is bicrucial with respect to squares if it is square-free but any extension of it to the right or to the left by any element gives a permutation that is not square-free. Avgustinovich et al. studied bicrucial permutations with respect to squares, and they proved that there exist bicrucial permutations of lengths 8k + 1, 8k + 5, 8k + 7 for k ≥ 1. It was left

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