Avgustinovich, Sergey and Kitaev, Sergey and Valyuzhenich, Alexander (2012) Crucial and bicrucial permutations with respect to arithmetic monotone patterns. Siberian Electronic Mathematical Reports, 9. pp. 660-671.Full text not available in this repository. Request a copy from the Strathclyde author
A pattern τ is a permutation, and an arithmetic occurrence of τ in (another) permutation π=π1π2...πn is a subsequence πi1πi2...πim of π that is order isomorphic to τ where the numbers i1<i2<...<im form an arithmetic progression. A permutation is (k,ℓ)-crucial if it avoids arithmetically the patterns 12...k and ℓ(ℓ−1)...1 but its extension to the right by any element does not avoid arithmetically these patterns. A (k,ℓ)-crucial permutation that cannot be extended to the left without creating an arithmetic occurrence of 12...k or ℓ(ℓ−1)...1 is called (k,ℓ)-bicrucial. In this paper we prove that arbitrary long (k,ℓ)-crucial and (k,ℓ)-bicrucial permutations exist for any k,ℓ≥3. Moreover, we show that the minimal length of a (k,ℓ)-crucial permutation is max(k,ℓ)(min(k,ℓ)−1), while the minimal length of a (k,ℓ)-bicrucial permutation is at most 2max(k,ℓ)(min(k,ℓ)−1), again for k,ℓ≥3.
|Keywords:||crucial permutation, bicrucial permutation, monotone pattern, arithmetic pattern, minimal length, Mathematics, Mathematics(all)|
|Subjects:||Science > Mathematics|
|Department:||Faculty of Science > Computer and Information Sciences|
|Depositing user:||Pure Administrator|
|Date Deposited:||17 Oct 2014 14:06|
|Last modified:||07 Jan 2017 04:21|