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For a positive integer $k$, define $S_{n,k}$ to be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles, i.e., \\[ S_{n,k} = \\{\\pi \\in S_{n}: \\pi = c_{1}c_{2} \\cdots c_{k}\\},\\] where $c_1,c_2,\\dots ,c_k$ are disjoint cycles. The size of $S_{n,k}$ is given by $\\left [ \\begin{matrix}n\\\\ k \\end{matrix}\\right]=(-1)^{n-k}s(n,k)$, where $s(n,k)$ is the Stirling number of the first kind. A family $\\mathcal{A} \\subseteq S_{n,k}$ is said to be $t$-{\\em intersecting} if any two elements of $\\mathcal{A}$ have at least $"},"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":"1402.0668","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-04T09:26:47Z","cross_cats_sorted":[],"title_canon_sha256":"72400d4ad4464fcd720fca10cb5442fbf97a7fe20053c4ccb037344e43e95109","abstract_canon_sha256":"4ac3bd3545a9999f89340692c11a650c32fd9e4fdf5a8496844c3c646ba1a9f6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:13.353684Z","signature_b64":"6ccFEjIDE4FylAETVJBrfF2wuaktxIB7iSB0SknwIsuTNJZBgSv+/vowSzFTeGVKiV5a36jZzExgiV354dfNCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb7b1ebab4933be33e3785f0835e2d975d99a42d67f611ce6cee5e5aa46e59dc","last_reissued_at":"2026-05-18T03:00:13.353100Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:13.353100Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Erd{\\H o}s-Ko-Rado theorem for permutations with fixed number of cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheng Yeaw Ku, Kok Bin Wong","submitted_at":"2014-02-04T09:26:47Z","abstract_excerpt":"Let $S_{n}$ denote the set of permutations of $[n]=\\{1,2,\\dots, n\\}$. For a positive integer $k$, define $S_{n,k}$ to be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles, i.e., \\[ S_{n,k} = \\{\\pi \\in S_{n}: \\pi = c_{1}c_{2} \\cdots c_{k}\\},\\] where $c_1,c_2,\\dots ,c_k$ are disjoint cycles. The size of $S_{n,k}$ is given by $\\left [ \\begin{matrix}n\\\\ k \\end{matrix}\\right]=(-1)^{n-k}s(n,k)$, where $s(n,k)$ is the Stirling number of the first kind. A family $\\mathcal{A} \\subseteq S_{n,k}$ is said to be $t$-{\\em intersecting} if any two elements of $\\mathcal{A}$ have at least $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0668","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":"1402.0668","created_at":"2026-05-18T03:00:13.353193+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.0668v1","created_at":"2026-05-18T03:00:13.353193+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.0668","created_at":"2026-05-18T03:00:13.353193+00:00"},{"alias_kind":"pith_short_12","alias_value":"7N5R5OVUSM56","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"7N5R5OVUSM56GPRX","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"7N5R5OVU","created_at":"2026-05-18T12:28:19.803747+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/7N5R5OVUSM56GPRXQXYIGXRNS5","json":"https://pith.science/pith/7N5R5OVUSM56GPRXQXYIGXRNS5.json","graph_json":"https://pith.science/api/pith-number/7N5R5OVUSM56GPRXQXYIGXRNS5/graph.json","events_json":"https://pith.science/api/pith-number/7N5R5OVUSM56GPRXQXYIGXRNS5/events.json","paper":"https://pith.science/paper/7N5R5OVU"},"agent_actions":{"view_html":"https://pith.science/pith/7N5R5OVUSM56GPRXQXYIGXRNS5","download_json":"https://pith.science/pith/7N5R5OVUSM56GPRXQXYIGXRNS5.json","view_paper":"https://pith.science/paper/7N5R5OVU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.0668&json=true","fetch_graph":"https://pith.science/api/pith-number/7N5R5OVUSM56GPRXQXYIGXRNS5/graph.json","fetch_events":"https://pith.science/api/pith-number/7N5R5OVUSM56GPRXQXYIGXRNS5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7N5R5OVUSM56GPRXQXYIGXRNS5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7N5R5OVUSM56GPRXQXYIGXRNS5/action/storage_attestation","attest_author":"https://pith.science/pith/7N5R5OVUSM56GPRXQXYIGXRNS5/action/author_attestation","sign_citation":"https://pith.science/pith/7N5R5OVUSM56GPRXQXYIGXRNS5/action/citation_signature","submit_replication":"https://pith.science/pith/7N5R5OVUSM56GPRXQXYIGXRNS5/action/replication_record"}},"created_at":"2026-05-18T03:00:13.353193+00:00","updated_at":"2026-05-18T03:00:13.353193+00:00"}