{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:J7IELOJGVAN72CZLOX53I7XPVA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4fb2a9a300ded856ae150c3446296988af3c45bd7faf9b6c97232e7b5b393f3d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-28T15:22:00Z","title_canon_sha256":"6594adb476a9f1bc9458527072f83d741c2af84b2cd28f6232ca955eb5c65892"},"schema_version":"1.0","source":{"id":"1211.6630","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.6630","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"arxiv_version","alias_value":"1211.6630v3","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6630","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"pith_short_12","alias_value":"J7IELOJGVAN7","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"J7IELOJGVAN72CZL","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"J7IELOJG","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:6241f11e052c4d994d8adcd98e9d3548d3b3238c862e42fb1a896a26dc9e2b1a","target":"graph","created_at":"2026-05-18T01:30:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We present various results on multiplying cycles in the symmetric group. Our first result is a generalisation of the following theorem of Boccara (1980): the number of ways of writing an odd permutation in the symmetric group on $n$ symbols as a product of an $n$-cycle and an $n-1$-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths.","authors_text":"Amarpreet Rattan, Valentin F\\'eray","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-28T15:22:00Z","title":"On products of long cycles: short cycle dependence and separation probabilities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6630","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5b09b10fdb461f86f0c17744d663750384bf33f2cd061fe7857a27625101134d","target":"record","created_at":"2026-05-18T01:30:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4fb2a9a300ded856ae150c3446296988af3c45bd7faf9b6c97232e7b5b393f3d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-28T15:22:00Z","title_canon_sha256":"6594adb476a9f1bc9458527072f83d741c2af84b2cd28f6232ca955eb5c65892"},"schema_version":"1.0","source":{"id":"1211.6630","kind":"arxiv","version":3}},"canonical_sha256":"4fd045b926a81bfd0b2b75fbb47eefa82a00ce21ef63778ca421acf044d686f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4fd045b926a81bfd0b2b75fbb47eefa82a00ce21ef63778ca421acf044d686f1","first_computed_at":"2026-05-18T01:30:41.329509Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:41.329509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pNtfVLjX3pAQoNu+qR6ifJvYcs0ie4qvZdCtktDwDJkyMHqyC5PF0da0X6bFRcvk/vMHEh9OSoHUchR14yslAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:41.329999Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.6630","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b09b10fdb461f86f0c17744d663750384bf33f2cd061fe7857a27625101134d","sha256:6241f11e052c4d994d8adcd98e9d3548d3b3238c862e42fb1a896a26dc9e2b1a"],"state_sha256":"3c59840000b8a25508980a184501c570ac6a1fc4fa1500260b161a24e984ccfc"}