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We give three different proofs of this fact using, respectively, an enumeration relying on the inclusion-exclusion principle, the introduction of two different Markov chains to generate uniform random permutations, and the construction of a combinatorial bijection. We also obtain the distribution of the analogous set for circular permutations that consists of those k in [n] such that \\Pi(k+1 mod "},"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":"1308.5459","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-25T22:40:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9449dd8d01cb1c65bec937c1f847e9f27f50306bcb3688bd47755d1c50de1e3a","abstract_canon_sha256":"490e0035619b6fc97ee2342eab824a76ace75156d3d82a29eb9c3cc63fcde79c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:08.140590Z","signature_b64":"R/i/xeSwi8OtjWoJh/GBv3Qi2MEN8xq3NdyYEE6XYxMhk1hsR4we7OiU53LBhwqIIURIuJyHdw540YT4ZXFVDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b83ca9a758af517efea608cda68e619df2d8c0b126731cc1c670bbeaa3ebb9df","last_reissued_at":"2026-05-18T02:53:08.139928Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:08.139928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unseparated pairs and fixed points in random permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Persi Diaconis, Ron Graham, Steven N. 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