Scrucial and bicrucial permutations with respect to squares
Gent, Ian and Kitaev, Sergey and Konovalov, Alexander and Linton, Steve and Nightingale, Peter (2015) Scrucial and bicrucial permutations with respect to squares. Journal of Integer Sequences, 18.

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Abstract
A permutation is squarefree if it does not contain two consecutive factors of length two or more that are orderisomorphic. A permutation is bicrucial with respect to squares if it is squarefree but any extension of it to the right or to the left by any element gives a permutation that is not squarefree. 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 leftcrucial (that is, not extendable to the left) squarefree 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 rightcrucial, leftcrucial, 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 Scrucial 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 Scrucial 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 squarefree permutations of length 19.
Item type:  Article 

ID code:  53717 
Keywords:  crucial permutation, Scrucial permutation, Pcrucial permutation, square, bicrucial permutation, Electronic computers. Computer science, Computational Theory and Mathematics 
Subjects:  Science > Mathematics > Electronic computers. Computer science 
Department:  Faculty of Science > Computer and Information Sciences 
Depositing user:  Pure Administrator 
Date deposited:  14 Jul 2015 08:29 
Last modified:  01 Apr 2017 04:57 
Related URLs:  
URI:  http://strathprints.strath.ac.uk/id/eprint/53717 
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