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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 $\\mathrm{s}^+$","authors_text":"Daniel Daly, Petr Vojt\\v{e}chovsk\\'y","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T15:05:14Z","title":"How permutations displace points and stretch intervals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05649","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:777c1e4e673eb156b788e1920573d56a2693c3a6d2e3d98b3ec15fe139964826","target":"record","created_at":"2026-05-18T01:32:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6a503aec891f483481ec4a97364742e5fe15bdeba45db40f0343c421a24d263f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T15:05:14Z","title_canon_sha256":"1bdd5a7e046d57bb5c36157362491ba396de0a15126eef4a2a010599ef5549ec"},"schema_version":"1.0","source":{"id":"1509.05649","kind":"arxiv","version":1}},"canonical_sha256":"3a65a623c096ed29c2193e85f76df69431e5dfc87d6736d09974e08f51f0d16b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a65a623c096ed29c2193e85f76df69431e5dfc87d6736d09974e08f51f0d16b","first_computed_at":"2026-05-18T01:32:42.692857Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:42.692857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2PW2Tg9URuJeu0W3DR9lYTEoTBFWx5Nwwd43wtWwiZGvbKIAgPEcZrb5GVaGcw7PlD9wq69y8UVWCJ9tMWzRDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:42.693370Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.05649","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:777c1e4e673eb156b788e1920573d56a2693c3a6d2e3d98b3ec15fe139964826","sha256:4453826c57f600ff6e697e1c18a5cba1d610b91d43b61bc06104ed44cbcbe933"],"state_sha256":"8db04106a3de54186981f9a07be4b7021a13b7e815c409fc2d1e18611bff9935"}