{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:M36FGJLR5ONGOXMWKNBMJKDBU7","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":"8a30fe8a159a01c73b80f71987d9420d939304615fd4394b3a546c5ff47f6334","cross_cats_sorted":["math.CO","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-10-08T20:46:09Z","title_canon_sha256":"9fee71acda2d8ac623d90fd950c6691ff0e95e94ee1cd834e017d575234f5bfd"},"schema_version":"1.0","source":{"id":"0910.1610","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.1610","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"0910.1610v4","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.1610","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"M36FGJLR5ONG","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"M36FGJLR5ONGOXMW","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"M36FGJLR","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:77799b59982ae6ce3ef68489ef224ef2dc714d232e350b3e2992f4ae969b766e","target":"graph","created_at":"2026-05-18T03:26:32Z","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":"There are two seemingly unrelated ideals associated with a simplicial complex \\Delta. One is the Stanley-Reisner ideal I_\\Delta, the monomial ideal generated by minimal non-faces of \\Delta, well-known in combinatorial commutative algebra. The other is the toric ideal I_{M(\\Delta)} of the facet subring of \\Delta, whose generators give a Markov basis for the hierarchical model defined by \\Delta, playing a prominent role in algebraic statistics.\n  In this note we show that the complexity of the generators of I_{M(\\Delta)} is determined by the Betti numbers of I_\\Delta. The unexpected connection b","authors_text":"Erik Stokes, Sonja Petrovi\\'c","cross_cats":["math.CO","math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-10-08T20:46:09Z","title":"Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.1610","kind":"arxiv","version":4},"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":"sha256:0c5e2c17a4ee5dd12f78d40fb4e2b76960ce1d1bb7d35be7b69dc96412e582bd","target":"record","created_at":"2026-05-18T03:26:32Z","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":"8a30fe8a159a01c73b80f71987d9420d939304615fd4394b3a546c5ff47f6334","cross_cats_sorted":["math.CO","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-10-08T20:46:09Z","title_canon_sha256":"9fee71acda2d8ac623d90fd950c6691ff0e95e94ee1cd834e017d575234f5bfd"},"schema_version":"1.0","source":{"id":"0910.1610","kind":"arxiv","version":4}},"canonical_sha256":"66fc532571eb9a675d965342c4a861a7e609f7bf3c20150de5b2dc8fcb7d8eb0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66fc532571eb9a675d965342c4a861a7e609f7bf3c20150de5b2dc8fcb7d8eb0","first_computed_at":"2026-05-18T03:26:32.794342Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:32.794342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IZnlAT2CrUMWYxHGvNtcJO9aZjVGd8yjl0IxZiC4KfBSfcU88ofb0DWdRk906sYAWBYFNvMKsLC6zkOyRieGBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:32.794693Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.1610","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0c5e2c17a4ee5dd12f78d40fb4e2b76960ce1d1bb7d35be7b69dc96412e582bd","sha256:77799b59982ae6ce3ef68489ef224ef2dc714d232e350b3e2992f4ae969b766e"],"state_sha256":"5c456d036726268781f35358700a960d048254b23927b5ee2d92b33a1150a550"}