{"paper":{"title":"Permutations with few inversions are locally uniform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Bevan","submitted_at":"2019-08-20T11:28:34Z","abstract_excerpt":"We prove that permutations with few inversions exhibit a local-global dichotomy in the following sense. Suppose ${\\boldsymbol\\sigma}$ is a permutation chosen uniformly at random from the set of all permutations of $[n]$ with exactly $m=m(n)\\ll n^2$ inversions. If $i<j$ are chosen uniformly at random from $[n]$, then ${\\boldsymbol\\sigma}(i)<{\\boldsymbol\\sigma}(j)$ asymptotically almost surely. However, if $i$ and $j$ are chosen so that $j-i\\ll m/n$, and $m \\ll n^2/\\log^2 n$, then $\\lim_{n\\to\\infty}\\mathbb{P}\\big[{\\boldsymbol\\sigma}(i)<{\\boldsymbol\\sigma}(j)\\big]=\\frac{1}{2}$. Moreover, if $k=k("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.07277","kind":"arxiv","version":2},"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/1908.07277/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"}