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A permutation is square-free if it does not contain two consecutive factorsof length two or more that are order-isomorphic. A permutation is bicrucialwith respect to squares if it is square-free but any extension of it to theright or to the left by any element gives a permutation that is notsquare-free.  Bicrucial permutations with respect to squares were studied by Avgustinovichet al., who 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 bicrucialpermutations of even length, or such permutations of length $8k+3$ exist. Inthis paper, we provide an encoding of orderings which allows us, using theconstraint solver Minion, to show that bicrucial permutations of even lengthexist, and the smallest such permutations are of length 32. To show that 32 isthe minimum length in question, we establish a result on left-crucial (that is,not extendable to the left) square-free permutations which begin with threeelements 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 generalise the notions of right-crucial, left-crucial, andbicrucial permutations studied in the literature in various contexts, byintroducing the notion of $P$-crucial permutations that can be extended to thenotion 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 thatS-crucial permutations exist with respect to squares, and minimal suchpermutations are of length 17.  Finally, using our software, we generate much of relevant data showing, forexample, that there are 162,190,472 bicrucial square-free permutations oflength 19.

S-crucial and bicrucial permutations with respect to squares