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We show the following lower and upper bounds in the case n >= k: F(n,k) \\in n^k k^{-k/2 + O(k/log k)}. As a consequence of the lower bound, a set of n-permutations each two having a reverse has size at most n^{n/2 + O(n/log n)}."},"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":"1208.2847","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-14T12:38:57Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"f8e019e25a832db03af7e3e7518924a8c22185f7624ccad578d4e887da8d9ad0","abstract_canon_sha256":"03c5769192f63b0ea3f5313f545ec6c8d6fd0f8890a881bbdcf37e1530feff0b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:41.326265Z","signature_b64":"KYwPiEjsl9sq1M6Pt4SW3bSC7b1uP/nUemugSv3r9sz5PdW12X6A2yWRf7Vs2jVagZKaEI8zpwFhAx+Vd6Y6CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04ae01c9766f2673ea3f4256fd1a438eb5a71c3a726a9c604987884b8d90189b","last_reissued_at":"2026-05-18T03:07:41.325730Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:41.325730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximum size of reverse-free sets of permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Josef Cibulka","submitted_at":"2012-08-14T12:38:57Z","abstract_excerpt":"Two words have a reverse if they have the same pair of distinct letters on the same pair of positions, but in reversed order. A set of words no two of which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size of a reverse-free set of words from [n]^k where no letter repeats within a word. We show the following lower and upper bounds in the case n >= k: F(n,k) \\in n^k k^{-k/2 + O(k/log k)}. 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