{"paper":{"title":"Crucial and bicrucial permutations with respect to arithmetic monotone patterns","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Valyuzhenich, Sergey Avgustinovich, Sergey Kitaev","submitted_at":"2012-10-09T14:51:57Z","abstract_excerpt":"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. A permutation 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 element does not avoid arithmetically these patterns. A $(k,\\ell)$-crucial permutation that cannot be extended to the left without creating an arithmetic occurrence "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2621","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}