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S-crucial and bicrucial permutations with respect to squares [article]

Ian Gent, Sergey Kitaev, Alexander Konovalov, Steve Linton, Peter Nightingale
2014 arXiv   pre-print
We show that S-crucial permutations exist with respect to squares, and minimal such permutations are of length 17.  ...  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.  ...  Thus, for any P , P -crucial permutations with respect to squares are equivalent to S-crucial permutations with respect to squares for some S ⊆ P .  ...

S-Crucial and Bicrucial Permutations with Respect to Squares

Ian Gent, Sergey Kitaev, Alexander Konovalov, Steve Linton, Peter Nightingale
2015 Journal of Integer Sequences   unpublished
We show that S-crucial permutations exist with respect to squares, and minimal such permutations are of length 17.  ...  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.  ...  S-crucial permutations with respect to squares exist, and the shortest such permutation is of length 17. There are 1568 S-crucial permutations with respect to squares of length 17. Proof.  ...

The lengths for which bicrucial square-free permutations exist [article]

Carla Groenland, Tom Johnston
2021 arXiv   pre-print
A square is a factor S = (S_1; S_2) where S_1 and S_2 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.  ...  The definition of bicrucial with respect to squares easily extends to bicrucial with respect to containing a kth power.  ...

СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ CRUCIAL AND BICRUCIAL PERMUTATIONS WITH RESPECT TO ARITHMETIC MONOTONE PATTERNS

S Avgustinovich, S Kitaev, A Valyuzhenich
unpublished
A pattern τ is a permutation, and an arithmetic occurrence of τ in (another) permutation π = π1π2. .  ...  Bicrucial words are called maximal words in the literature and they were studied with respect to so-called squares and abelian squares [6, 14] .  ...  (Bi)crucial permutations with respect to squares were studied in  , while the present paper deals with arithmetic occurrences of monotone patterns in permutations and (bi)crucial permutations associated  ...

Solving computational problems in the theory of word-representable graphs [article]

Özgür Akgün, Ian P. Gent, Sergey Kitaev, Hans Zantema
2018 arXiv   pre-print
In particular, we find a new graph on 9 vertices with high representation number.  ...  Finally, we introduce the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation, and show computationally that the refinement is not equivalent to the original  ...  permutations with respect to squares  .  ...