{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:AFCZHD2LDHOK4BA6NMO6TMU4MD","short_pith_number":"pith:AFCZHD2L","schema_version":"1.0","canonical_sha256":"0145938f4b19dcae041e6b1de9b29c60ddc95d13fdbc05c2f71836921e299882","source":{"kind":"arxiv","id":"1605.01068","version":3},"attestation_state":"computed","paper":{"title":"Permutations contained in transitive subgroups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Dimitris Koukoulopoulos, Kevin Ford, Sean Eberhard","submitted_at":"2016-05-03T20:02:14Z","abstract_excerpt":"In the first paper in this series we estimated the probability that a random permutation $\\pi\\in\\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $\\pi$ has $m$ disjoint fixed sets of prescribed sizes $k_1,\\dots,k_m$, where $k_1+\\cdots+k_m=n$. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than $\\mathcal{S}_n$ or $\\mathcal{A}_n$. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an e"},"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":"1605.01068","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2016-05-03T20:02:14Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5f89183ae1c52c872e6d8d776381b7910f357d4dbed4042ccf5e492c21241543","abstract_canon_sha256":"ce1403b42577ec065fe6ca9b5772b89655dfc000633b93bb0808667924813e38"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:44.431962Z","signature_b64":"Ahv13L3Xh3gBinxzfl0pGE2IC9MRBI5cTKSpWyd/ZIBAGH1yeZe5cO7TbWfCDAr3EqRyGZ7xET/yHtDAeo4RBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0145938f4b19dcae041e6b1de9b29c60ddc95d13fdbc05c2f71836921e299882","last_reissued_at":"2026-05-18T00:42:44.431383Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:44.431383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Permutations contained in transitive subgroups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Dimitris Koukoulopoulos, Kevin Ford, Sean Eberhard","submitted_at":"2016-05-03T20:02:14Z","abstract_excerpt":"In the first paper in this series we estimated the probability that a random permutation $\\pi\\in\\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $\\pi$ has $m$ disjoint fixed sets of prescribed sizes $k_1,\\dots,k_m$, where $k_1+\\cdots+k_m=n$. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than $\\mathcal{S}_n$ or $\\mathcal{A}_n$. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01068","kind":"arxiv","version":3},"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":"1605.01068","created_at":"2026-05-18T00:42:44.431474+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.01068v3","created_at":"2026-05-18T00:42:44.431474+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.01068","created_at":"2026-05-18T00:42:44.431474+00:00"},{"alias_kind":"pith_short_12","alias_value":"AFCZHD2LDHOK","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"AFCZHD2LDHOK4BA6","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"AFCZHD2L","created_at":"2026-05-18T12:30:07.202191+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/AFCZHD2LDHOK4BA6NMO6TMU4MD","json":"https://pith.science/pith/AFCZHD2LDHOK4BA6NMO6TMU4MD.json","graph_json":"https://pith.science/api/pith-number/AFCZHD2LDHOK4BA6NMO6TMU4MD/graph.json","events_json":"https://pith.science/api/pith-number/AFCZHD2LDHOK4BA6NMO6TMU4MD/events.json","paper":"https://pith.science/paper/AFCZHD2L"},"agent_actions":{"view_html":"https://pith.science/pith/AFCZHD2LDHOK4BA6NMO6TMU4MD","download_json":"https://pith.science/pith/AFCZHD2LDHOK4BA6NMO6TMU4MD.json","view_paper":"https://pith.science/paper/AFCZHD2L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.01068&json=true","fetch_graph":"https://pith.science/api/pith-number/AFCZHD2LDHOK4BA6NMO6TMU4MD/graph.json","fetch_events":"https://pith.science/api/pith-number/AFCZHD2LDHOK4BA6NMO6TMU4MD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AFCZHD2LDHOK4BA6NMO6TMU4MD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AFCZHD2LDHOK4BA6NMO6TMU4MD/action/storage_attestation","attest_author":"https://pith.science/pith/AFCZHD2LDHOK4BA6NMO6TMU4MD/action/author_attestation","sign_citation":"https://pith.science/pith/AFCZHD2LDHOK4BA6NMO6TMU4MD/action/citation_signature","submit_replication":"https://pith.science/pith/AFCZHD2LDHOK4BA6NMO6TMU4MD/action/replication_record"}},"created_at":"2026-05-18T00:42:44.431474+00:00","updated_at":"2026-05-18T00:42:44.431474+00:00"}