{"paper":{"title":"A proof of the peak polynomial positivity conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Diaz-Lopez, Erik Insko, Mohamed Omar, Pamela E. Harris","submitted_at":"2016-05-05T19:48:50Z","abstract_excerpt":"We say that a permutation $\\pi=\\pi_1\\pi_2\\cdots \\pi_n \\in \\mathfrak{S}_n$ has a peak at index $i$ if $\\pi_{i-1} < \\pi_i > \\pi_{i+1}$. Let $\\mathcal{P}(\\pi)$ denote the set of indices where $\\pi$ has a peak. Given a set $S$ of positive integers, we define $\\mathcal{P}_S(n)=\\{\\pi\\in\\mathfrak{S}_n:\\mathcal{P}(\\pi)=S\\}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| \\mathcal{P}_S(n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01708","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"}