S-Crucial and Bicrucial Permutations with Respect to Squares

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 $k\ge 1$. It was left as open questions whether bicrucial permutations of even length, or such permutations of length $8k+3$ exist. In this paper, we provide an encoding of orderings which allows us, using the constraint solver Minion, to show that bicrucial permutations of even length exist, and the smallest such permutations are of length 32. To show that 32 is the minimum length in question, we establish a result on left-crucial (that is, not extendable to the left) square-free permutations which begin with three elements in monotone order. Also, we show that bicrucial permutations of length $8k+3$ exist for $k=2,3$ and they do not exist for $k=1$. Further, we generalize the notions of right-crucial, left-crucial, and bicrucial permutations studied in the literature in various contexts, by introducing the notion of $P$-crucial permutations that can be extended to the notion of $P$-crucial words. In S-crucial permutations, a particular case of $P$-crucial permutations, we deal with permutations that avoid prohibitions, but whose extensions in any position contain a prohibition. We show that S-crucial permutations exist with respect to squares, and minimal such permutations are of length 17. Finally, using our software, we generate relevant data showing, for example, that there are 162,190,472 bicrucial square-free permutations of length 19.