{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:CBBRDAZBBDMLCG5L3YZ3Z7WFRQ","short_pith_number":"pith:CBBRDAZB","schema_version":"1.0","canonical_sha256":"104311832108d8b11babde33bcfec58c0d6d86c992e687266c57977933bc241a","source":{"kind":"arxiv","id":"math/0603554","version":1},"attestation_state":"computed","paper":{"title":"On the frequency of permutations containing a long cycle","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Alice C. Niemeyer, Cheryl E. Praeger","submitted_at":"2006-03-23T05:47:18Z","abstract_excerpt":"A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \\le m \\le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the conditional probabilities that an element of $S_n$ or $A_n$ contains an $r$-cycle, given that it satisfies an equation of the form $x^{rs}=1$ where $s\\leq3$. For example, the conditional probability that an element $x$ is an $n$-cycle, given that $x^n=1$, is always greater than 2/7, and is greater than 1/2 if $n$ does not divide 24. Our results improve estimates of"},"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":"math/0603554","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2006-03-23T05:47:18Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"57aefa8626123834ca0879d511c6738d9f8e602cb83390d630ebfe342fbb89bc","abstract_canon_sha256":"ca4150e864d27caf481d284a84441506d82bcd838092515996a8fb2fd1a3337b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:48.060570Z","signature_b64":"Qre4bE+LxxjypmksQS6biTY/8ldw3JCKKG2/4oKIvmRIFvD6qkYRkzbqWJIrCj7f8hRkozXSuoTaWeLkyQOjDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"104311832108d8b11babde33bcfec58c0d6d86c992e687266c57977933bc241a","last_reissued_at":"2026-05-18T02:52:48.060190Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:48.060190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the frequency of permutations containing a long cycle","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Alice C. Niemeyer, Cheryl E. Praeger","submitted_at":"2006-03-23T05:47:18Z","abstract_excerpt":"A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \\le m \\le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the conditional probabilities that an element of $S_n$ or $A_n$ contains an $r$-cycle, given that it satisfies an equation of the form $x^{rs}=1$ where $s\\leq3$. For example, the conditional probability that an element $x$ is an $n$-cycle, given that $x^n=1$, is always greater than 2/7, and is greater than 1/2 if $n$ does not divide 24. Our results improve estimates of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603554","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":"math/0603554","created_at":"2026-05-18T02:52:48.060245+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0603554v1","created_at":"2026-05-18T02:52:48.060245+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0603554","created_at":"2026-05-18T02:52:48.060245+00:00"},{"alias_kind":"pith_short_12","alias_value":"CBBRDAZBBDML","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"CBBRDAZBBDMLCG5L","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"CBBRDAZB","created_at":"2026-05-18T12:25:53.939244+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/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ","json":"https://pith.science/pith/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ.json","graph_json":"https://pith.science/api/pith-number/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/graph.json","events_json":"https://pith.science/api/pith-number/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/events.json","paper":"https://pith.science/paper/CBBRDAZB"},"agent_actions":{"view_html":"https://pith.science/pith/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ","download_json":"https://pith.science/pith/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ.json","view_paper":"https://pith.science/paper/CBBRDAZB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0603554&json=true","fetch_graph":"https://pith.science/api/pith-number/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/graph.json","fetch_events":"https://pith.science/api/pith-number/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/action/storage_attestation","attest_author":"https://pith.science/pith/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/action/author_attestation","sign_citation":"https://pith.science/pith/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/action/citation_signature","submit_replication":"https://pith.science/pith/CBBRDAZBBDMLCG5L3YZ3Z7WFRQ/action/replication_record"}},"created_at":"2026-05-18T02:52:48.060245+00:00","updated_at":"2026-05-18T02:52:48.060245+00:00"}