{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:VKOIS24W45LWQZCXS3YDGSE5YO","short_pith_number":"pith:VKOIS24W","canonical_record":{"source":{"id":"0808.2307","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-08-17T19:06:47Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"cde809113bd8bce01baed5de72f419cb5fddddb90bea9637c85de8680a1cb8a1","abstract_canon_sha256":"dd41d948ed84ed570c3b11e384b2e9376d9fde2dedabd1e79825f9016b3971aa"},"schema_version":"1.0"},"canonical_sha256":"aa9c896b96e75768645796f033489dc3a0ee9ed3c7bc81a0a85b99c9f4395877","source":{"kind":"arxiv","id":"0808.2307","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0808.2307","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"arxiv_version","alias_value":"0808.2307v3","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.2307","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"pith_short_12","alias_value":"VKOIS24W45LW","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"VKOIS24W45LWQZCX","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"VKOIS24W","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:VKOIS24W45LWQZCXS3YDGSE5YO","target":"record","payload":{"canonical_record":{"source":{"id":"0808.2307","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-08-17T19:06:47Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"cde809113bd8bce01baed5de72f419cb5fddddb90bea9637c85de8680a1cb8a1","abstract_canon_sha256":"dd41d948ed84ed570c3b11e384b2e9376d9fde2dedabd1e79825f9016b3971aa"},"schema_version":"1.0"},"canonical_sha256":"aa9c896b96e75768645796f033489dc3a0ee9ed3c7bc81a0a85b99c9f4395877","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:22.260627Z","signature_b64":"GZaE4XtH0uqTT3AekGU7fRvO/2w6jnfk8lzYDaFAkD1tq9AVzvY7PYKTtr3PqdZVrnbkI/eXokjdBJhbVR75BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa9c896b96e75768645796f033489dc3a0ee9ed3c7bc81a0a85b99c9f4395877","last_reissued_at":"2026-05-18T04:42:22.259962Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:22.259962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0808.2307","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:42:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7i+OqzcFofsnOSahZn/SYSbFTgiegRA5xf2bkWItM3gkK2h2FBjztcwXEfZmPd8TrbIIcodQ2vBqGdDO6yliCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T18:12:31.977269Z"},"content_sha256":"ec5c9aa99f8601513f027088f5c446570ed02919825dbbd8d29757402115eef1","schema_version":"1.0","event_id":"sha256:ec5c9aa99f8601513f027088f5c446570ed02919825dbbd8d29757402115eef1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:VKOIS24W45LWQZCXS3YDGSE5YO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Boolean complexes for Ferrers graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Anders Claesson, Bridget Eileen Tenner, Kari Ragnarsson, Sergey Kitaev","submitted_at":"2008-08-17T19:06:47Z","abstract_excerpt":"In this paper we provide an explicit formula for calculating the boolean number of a Ferrers graph. By previous work of the last two authors, this determines the homotopy type of the boolean complex of the graph. Specializing to staircase shapes, we show that the boolean numbers of the associated Ferrers graphs are the Genocchi numbers of the second kind, and obtain a relation between the Legendre-Stirling numbers and the Genocchi numbers of the second kind. In another application, we compute the boolean number of a complete bipartite graph, corresponding to a rectangular Ferrers shape, which "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.2307","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:42:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d8Wz6aB6Hrg4ttqwIrKWSAxvmXb7fNlVPoJPyTXBvlnGsRhuJe3wCnPEyMhipOR3aCwcql0cQ1vNV2SaIxbpBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T18:12:31.977687Z"},"content_sha256":"3bb8d855440af735444e0b4e85f154648832f54e4a035f517f6dd7a85b685080","schema_version":"1.0","event_id":"sha256:3bb8d855440af735444e0b4e85f154648832f54e4a035f517f6dd7a85b685080"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VKOIS24W45LWQZCXS3YDGSE5YO/bundle.json","state_url":"https://pith.science/pith/VKOIS24W45LWQZCXS3YDGSE5YO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VKOIS24W45LWQZCXS3YDGSE5YO/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T18:12:31Z","links":{"resolver":"https://pith.science/pith/VKOIS24W45LWQZCXS3YDGSE5YO","bundle":"https://pith.science/pith/VKOIS24W45LWQZCXS3YDGSE5YO/bundle.json","state":"https://pith.science/pith/VKOIS24W45LWQZCXS3YDGSE5YO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VKOIS24W45LWQZCXS3YDGSE5YO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:VKOIS24W45LWQZCXS3YDGSE5YO","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":"dd41d948ed84ed570c3b11e384b2e9376d9fde2dedabd1e79825f9016b3971aa","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-08-17T19:06:47Z","title_canon_sha256":"cde809113bd8bce01baed5de72f419cb5fddddb90bea9637c85de8680a1cb8a1"},"schema_version":"1.0","source":{"id":"0808.2307","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0808.2307","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"arxiv_version","alias_value":"0808.2307v3","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.2307","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"pith_short_12","alias_value":"VKOIS24W45LW","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"VKOIS24W45LWQZCX","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"VKOIS24W","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:3bb8d855440af735444e0b4e85f154648832f54e4a035f517f6dd7a85b685080","target":"graph","created_at":"2026-05-18T04:42:22Z","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":"In this paper we provide an explicit formula for calculating the boolean number of a Ferrers graph. By previous work of the last two authors, this determines the homotopy type of the boolean complex of the graph. Specializing to staircase shapes, we show that the boolean numbers of the associated Ferrers graphs are the Genocchi numbers of the second kind, and obtain a relation between the Legendre-Stirling numbers and the Genocchi numbers of the second kind. In another application, we compute the boolean number of a complete bipartite graph, corresponding to a rectangular Ferrers shape, which ","authors_text":"Anders Claesson, Bridget Eileen Tenner, Kari Ragnarsson, Sergey Kitaev","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-08-17T19:06:47Z","title":"Boolean complexes for Ferrers graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.2307","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:ec5c9aa99f8601513f027088f5c446570ed02919825dbbd8d29757402115eef1","target":"record","created_at":"2026-05-18T04:42:22Z","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":"dd41d948ed84ed570c3b11e384b2e9376d9fde2dedabd1e79825f9016b3971aa","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-08-17T19:06:47Z","title_canon_sha256":"cde809113bd8bce01baed5de72f419cb5fddddb90bea9637c85de8680a1cb8a1"},"schema_version":"1.0","source":{"id":"0808.2307","kind":"arxiv","version":3}},"canonical_sha256":"aa9c896b96e75768645796f033489dc3a0ee9ed3c7bc81a0a85b99c9f4395877","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa9c896b96e75768645796f033489dc3a0ee9ed3c7bc81a0a85b99c9f4395877","first_computed_at":"2026-05-18T04:42:22.259962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:22.259962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GZaE4XtH0uqTT3AekGU7fRvO/2w6jnfk8lzYDaFAkD1tq9AVzvY7PYKTtr3PqdZVrnbkI/eXokjdBJhbVR75BA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:22.260627Z","signed_message":"canonical_sha256_bytes"},"source_id":"0808.2307","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec5c9aa99f8601513f027088f5c446570ed02919825dbbd8d29757402115eef1","sha256:3bb8d855440af735444e0b4e85f154648832f54e4a035f517f6dd7a85b685080"],"state_sha256":"622d69b058b847cf4a824f04393b759e3165371085543bc9c6701c74759124cf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nhjt/+7SorUg5y32YsTbx7zl5sj6uc7/fRZisq3z3jwvqLDvQ9/OrHMX6Eby28qFfLpFqyVjQnC+3DxX/MBOCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T18:12:31.979971Z","bundle_sha256":"6275dd279c18a3c048d02532fa34d71e2f3538cbb50fb38e8e0cfcf718b74438"}}