{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MVUUD5OCBZ5SJRIWE46RK3RGUS","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":"64c81d7d9ede4a729dce915557ce1c685a9b7ad69b653dde7fa12b5f076e50b2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-24T17:31:31Z","title_canon_sha256":"c2dc1b2c590126dde1297ea5d3f10934e54b2698c24866144103d694f7e00846"},"schema_version":"1.0","source":{"id":"1203.5433","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.5433","created_at":"2026-05-18T03:59:15Z"},{"alias_kind":"arxiv_version","alias_value":"1203.5433v1","created_at":"2026-05-18T03:59:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5433","created_at":"2026-05-18T03:59:15Z"},{"alias_kind":"pith_short_12","alias_value":"MVUUD5OCBZ5S","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MVUUD5OCBZ5SJRIW","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MVUUD5OC","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:bcbd015fa83fde4d25bdeaf1599124f03c9c4944a6ff44775c256ca366bf3903","target":"graph","created_at":"2026-05-18T03:59:15Z","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":"Let S_n be the set of all permutations on [n]:={1,2,....,n}. We denote by kappa_n the smallest cardinality of a subset A of S_{n+1} that \"covers\" S_n, in the sense that each pi in S_n may be found as an order-isomorphic subsequence of some pi' in A. What are general upper bounds on kappa_n? If we randomly select nu_n elements of S_{n+1}, when does the probability that they cover S_n transition from 0 to 1? Can we provide a fine-magnification analysis that provides the \"probability of coverage\" when nu_n is around the level given by the phase transition? In this paper we answer these questions ","authors_text":"Anant Godbole, Bill Kay, Kathryn Hawley, Taylor Allison","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-24T17:31:31Z","title":"Covering n-Permutations with (n+1)-Permutations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5433","kind":"arxiv","version":1},"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":"sha256:5483933ea98b754d1b345cc754865d0ee03fa9f751257476f5a85fd7ad170259","target":"record","created_at":"2026-05-18T03:59:15Z","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":"64c81d7d9ede4a729dce915557ce1c685a9b7ad69b653dde7fa12b5f076e50b2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-24T17:31:31Z","title_canon_sha256":"c2dc1b2c590126dde1297ea5d3f10934e54b2698c24866144103d694f7e00846"},"schema_version":"1.0","source":{"id":"1203.5433","kind":"arxiv","version":1}},"canonical_sha256":"656941f5c20e7b24c516273d156e26a49111de5617c7ea14e23e98f07c6846f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"656941f5c20e7b24c516273d156e26a49111de5617c7ea14e23e98f07c6846f2","first_computed_at":"2026-05-18T03:59:15.113191Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:15.113191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O4lWPkf31YqLiAW6gjiZ2nUnTL6DuLpEkVjKCj3EjMNLh0115y510Vsq+G0pmrS3hLaWmVBGS2xAVBxdEEyNDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:15.113795Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.5433","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5483933ea98b754d1b345cc754865d0ee03fa9f751257476f5a85fd7ad170259","sha256:bcbd015fa83fde4d25bdeaf1599124f03c9c4944a6ff44775c256ca366bf3903"],"state_sha256":"7e3b08a434c01a6483f1bd00cb84bf1d1e87705a12326cfc686b83a7dab99f11"}