{"paper":{"title":"How permutations displace points and stretch intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Daly, Petr Vojt\\v{e}chovsk\\'y","submitted_at":"2015-09-18T15:05:14Z","abstract_excerpt":"Let $S_n$ be the set of permutations on $\\{1,\\,\\dots,\\,n\\}$ and $\\pi\\in S_n$. Let $\\mathrm{d}(\\pi)$ be the arithmetic average of $\\{|i-\\pi(i)|;\\;1\\le i\\le n\\}$. Then $\\mathrm{d}(\\pi)/n\\in[0,\\,1/2]$, the expected value of $\\mathrm{d}(\\pi)/n$ approaches $1/3$ as $n$ approaches infinity, and $\\mathrm{d}(\\pi)/n$ is close to $1/3$ for most permutations. We describe all permutations $\\pi$ with maximal $\\mathrm{d}(\\pi)$.\n  Let $\\mathrm{s}^+(\\pi)$ and $\\mathrm{s}^*(\\pi)$ be the arithmetic and geometric averages of $\\{|\\pi(i)-\\pi(i+1)|;\\;1\\le i<n\\}$, and let $M^+$, $M^*$ be the maxima of $\\mathrm{s}^+$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05649","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"}