Picture of mobile phone running fintech app

Fintech: Open Access research exploring new frontiers in financial technology

Strathprints makes available Open Access scholarly outputs by the Department of Accounting & Finance at Strathclyde. Particular research specialisms include financial risk management and investment strategies.

The Department also hosts the Centre for Financial Regulation and Innovation (CeFRI), demonstrating research expertise in fintech and capital markets. It also aims to provide a strategic link between academia, policy-makers, regulators and other financial industry participants.

Explore all Strathclyde Open Access research...

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

Abstract

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.