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

2014
*
arXiv
*
pre-print

We show that

arXiv:1402.3582v1
fatcat:meybzjs6kbce3dn5zaen5eajwa
*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 . ...##
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S-Crucial and Bicrucial Permutations with Respect to Squares

2015
*
Journal of Integer Sequences
*
unpublished

We show that

fatcat:5wzbqnozrjhqhkig6mb34q4nqu
*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. ...##
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The lengths for which bicrucial square-free permutations exist
[article]

2021
*
arXiv
*
pre-print

A

arXiv:2109.00502v1
fatcat:u53yvcyyqze63mkaax3qadlfkm
*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. ...##
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СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ CRUCIAL AND BICRUCIAL PERMUTATIONS WITH RESPECT TO ARITHMETIC MONOTONE PATTERNS

unpublished

A pattern τ is a

fatcat:b2ithgklcrekfjhoirpbtxjgtm
*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 [4] , while the present paper deals*with*arithmetic occurrences of monotone patterns in*permutations**and*(bi)*crucial**permutations*associated ...##
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Solving computational problems in the theory of word-representable graphs
[article]

2018
*
arXiv
*
pre-print

In particular, we find a new graph on 9 vertices

arXiv:1808.01215v1
fatcat:yzl3vjtnajce3jxyi2g4nxwyna
*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*[9] . ...