{"paper":{"title":"What fraction of an $S_n$-orbit can lie on a hyperplane?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David McKinnon, Jiahui Huang, Matthew Satriano","submitted_at":"2020-01-24T18:14:10Z","abstract_excerpt":"Consider the $S_n$-action on $\\mathbb{R}^n$ given by permuting coordinates. This paper addresses the following problem: compute $\\max_{v,H} |H\\cap S_nv|$ as $H\\subset\\mathbb{R}^n$ ranges over all hyperplanes through the origin and $v\\in\\mathbb{R}^n$ ranges over all vectors with distinct coordinates that are not contained in the hyperplane $\\sum x_i=0$. We conjecture that for $n\\geq3$, the answer is $(n-1)!$ for odd $n$, and $n(n-2)!$ for even $n$. We prove that if $p$ is the largest prime with $p\\leq n$, then $\\max_{v,H} |H\\cap S_nv|\\leq \\frac{n!}{p}$. In particular, this proves the conjecture"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.09123","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2001.09123/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}