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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 $\\mathrm{s}^+$"},"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":"1509.05649","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T15:05:14Z","cross_cats_sorted":[],"title_canon_sha256":"1bdd5a7e046d57bb5c36157362491ba396de0a15126eef4a2a010599ef5549ec","abstract_canon_sha256":"6a503aec891f483481ec4a97364742e5fe15bdeba45db40f0343c421a24d263f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.693370Z","signature_b64":"2PW2Tg9URuJeu0W3DR9lYTEoTBFWx5Nwwd43wtWwiZGvbKIAgPEcZrb5GVaGcw7PlD9wq69y8UVWCJ9tMWzRDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a65a623c096ed29c2193e85f76df69431e5dfc87d6736d09974e08f51f0d16b","last_reissued_at":"2026-05-18T01:32:42.692857Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.692857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.05649","created_at":"2026-05-18T01:32:42.692957+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05649v1","created_at":"2026-05-18T01:32:42.692957+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05649","created_at":"2026-05-18T01:32:42.692957+00:00"},{"alias_kind":"pith_short_12","alias_value":"HJS2MI6AS3WS","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"HJS2MI6AS3WSTQQZ","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"HJS2MI6A","created_at":"2026-05-18T12:29:25.134429+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/HJS2MI6AS3WSTQQZH2C7O3PWSQ","json":"https://pith.science/pith/HJS2MI6AS3WSTQQZH2C7O3PWSQ.json","graph_json":"https://pith.science/api/pith-number/HJS2MI6AS3WSTQQZH2C7O3PWSQ/graph.json","events_json":"https://pith.science/api/pith-number/HJS2MI6AS3WSTQQZH2C7O3PWSQ/events.json","paper":"https://pith.science/paper/HJS2MI6A"},"agent_actions":{"view_html":"https://pith.science/pith/HJS2MI6AS3WSTQQZH2C7O3PWSQ","download_json":"https://pith.science/pith/HJS2MI6AS3WSTQQZH2C7O3PWSQ.json","view_paper":"https://pith.science/paper/HJS2MI6A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05649&json=true","fetch_graph":"https://pith.science/api/pith-number/HJS2MI6AS3WSTQQZH2C7O3PWSQ/graph.json","fetch_events":"https://pith.science/api/pith-number/HJS2MI6AS3WSTQQZH2C7O3PWSQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HJS2MI6AS3WSTQQZH2C7O3PWSQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HJS2MI6AS3WSTQQZH2C7O3PWSQ/action/storage_attestation","attest_author":"https://pith.science/pith/HJS2MI6AS3WSTQQZH2C7O3PWSQ/action/author_attestation","sign_citation":"https://pith.science/pith/HJS2MI6AS3WSTQQZH2C7O3PWSQ/action/citation_signature","submit_replication":"https://pith.science/pith/HJS2MI6AS3WSTQQZH2C7O3PWSQ/action/replication_record"}},"created_at":"2026-05-18T01:32:42.692957+00:00","updated_at":"2026-05-18T01:32:42.692957+00:00"}