1 research outputs found
Crucial and bicrucial permutations with respect to arithmetic monotone patterns
A pattern is a permutation, and an arithmetic occurrence of in
(another) permutation is a subsequence
of that is order isomorphic to
where the numbers form an arithmetic progression. A
permutation is -crucial if it avoids arithmetically the patterns
and but its extension to the right by any element
does not avoid arithmetically these patterns. A -crucial permutation
that cannot be extended to the left without creating an arithmetic occurrence
of or is called -bicrucial.
In this paper we prove that arbitrary long -crucial and
-bicrucial permutations exist for any . Moreover, we
show that the minimal length of a -crucial permutation is
, while the minimal length of a
-bicrucial permutation is at most ,
again for