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A pattern τ is a permutation, and an arithmetic occurrence of τ in (another) permutation π=π_1π_2...π_n is a subsequence π_i_1π_i_2...π_i_m of π that is order isomorphic to τ where the numbers i_1<i_2<...<i_m 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 leftarXiv:1210.2621v1 fatcat:qmkjgj54nrditcqjfevd3fzgdq