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.
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.
Original language | English |
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Pages (from-to) | 660-671 |
Number of pages | 12 |
Journal | Siberian Electronic Mathematical Reports |
Volume | 9 |
Publication status | Published - 19 Dec 2012 |
Keywords
- crucial permutation
- bicrucial permutation
- monotone pattern
- arithmetic pattern
- minimal length