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A pattern $\tau$ is a permutation, and an arithmetic occurrence of $\tau$ in(another) permutation $\pi=\pi_1\pi_2...\pi_n$ is a subsequence$\pi_{i_1}\pi_{i_2}...\pi_{i_m}$ of $\pi$ that is order isomorphic to $\tau$where the numbers $i_1<i_2<...<i_m$ form an arithmetic progression. Apermutation is $(k,\ell)$-crucial if it avoids arithmetically the patterns$12... k$ and $\ell(\ell-1)... 1$ but its extension to the right by any elementdoes not avoid arithmetically these patterns. A $(k,\ell)$-crucial permutationthat cannot be extended to the left without creating an arithmetic occurrenceof $12... k$ or $\ell(\ell-1)... 1$ is called $(k,\ell)$-bicrucial.  In this paper we prove that arbitrary long $(k,\ell)$-crucial and$(k,\ell)$-bicrucial permutations exist for any $k,\ell\geq 3$. Moreover, weshow that the minimal length of a $(k,\ell)$-crucial permutation is$\max(k,\ell)(\min(k,\ell)-1)$, while the minimal length of a$(k,\ell)$-bicrucial permutation is at most $2\max(k,\ell)(\min(k,\ell)-1)$,again for $k,\ell\geq3$.

Crucial and bicrucial permutations with respect to arithmetic monotone patterns