Picture of smart phone in human hand

World leading smartphone and mobile technology research at Strathclyde...

The Strathprints institutional repository is a digital archive of University of Strathclyde's Open Access research outputs. Strathprints provides access to thousands of Open Access research papers by University of Strathclyde researchers, including by Strathclyde researchers from the Department of Computer & Information Sciences involved in researching exciting new applications for mobile and smartphone technology. But the transformative application of mobile technologies is also the focus of research within disciplines as diverse as Electronic & Electrical Engineering, Marketing, Human Resource Management and Biomedical Enginering, among others.

Explore Strathclyde's Open Access research on smartphone technology now...

Crucial and bicrucial permutations with respect to arithmetic monotone patterns

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.