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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("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1908.07277","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-20T11:28:34Z","cross_cats_sorted":[],"title_canon_sha256":"a774da70c4ccdcda446a48a452e9ccfd70282ea4caebb97f002193094064bbdb","abstract_canon_sha256":"42734eca3e09753764ee6d1e16093c492103dd950dad26fff9075e105004c745"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:08:24.046994Z","signature_b64":"vaqyA+nv+kJwdngGwOxjIVsfJmo5LNWVQU86GhZ1YQZKid21ceaLQdlW4kcjHB8pK3jOU6MF50eTyOVU09IABg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9dedf25ce50ee765db57bad3f57bab9cc44a9b3c96c5df50ef7f7b79bf7e4da","last_reissued_at":"2026-07-05T05:08:24.046560Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:08:24.046560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.07277","created_at":"2026-07-05T05:08:24.046618+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.07277v2","created_at":"2026-07-05T05:08:24.046618+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.07277","created_at":"2026-07-05T05:08:24.046618+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZHPN6JOOKDXH","created_at":"2026-07-05T05:08:24.046618+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZHPN6JOOKDXHMXNV","created_at":"2026-07-05T05:08:24.046618+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZHPN6JOO","created_at":"2026-07-05T05:08:24.046618+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH","json":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH.json","graph_json":"https://pith.science/api/pith-number/ZHPN6JOOKDXHMXNVPOWT6V52XH/graph.json","events_json":"https://pith.science/api/pith-number/ZHPN6JOOKDXHMXNVPOWT6V52XH/events.json","paper":"https://pith.science/paper/ZHPN6JOO"},"agent_actions":{"view_html":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH","download_json":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH.json","view_paper":"https://pith.science/paper/ZHPN6JOO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.07277&json=true","fetch_graph":"https://pith.science/api/pith-number/ZHPN6JOOKDXHMXNVPOWT6V52XH/graph.json","fetch_events":"https://pith.science/api/pith-number/ZHPN6JOOKDXHMXNVPOWT6V52XH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH/action/storage_attestation","attest_author":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH/action/author_attestation","sign_citation":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH/action/citation_signature","submit_replication":"https://pith.science/pith/ZHPN6JOOKDXHMXNVPOWT6V52XH/action/replication_record"}},"created_at":"2026-07-05T05:08:24.046618+00:00","updated_at":"2026-07-05T05:08:24.046618+00:00"}