{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:NC7GTMGJCBJ3C4CB4QOGR3YP7M","short_pith_number":"pith:NC7GTMGJ","canonical_record":{"source":{"id":"0802.2576","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-02-19T02:04:49Z","cross_cats_sorted":[],"title_canon_sha256":"e79c77e598a1cba77fd4e920f773d4d322661bb8a8fa686c706e361c3214f7df","abstract_canon_sha256":"7a31f4835c66b4db52f72cb09135a4595ba453b27232d9bdce01c930be0eef43"},"schema_version":"1.0"},"canonical_sha256":"68be69b0c91053b17041e41c68ef0ffb3d900e66d1e8db5c8b8c92a9a6300b22","source":{"kind":"arxiv","id":"0802.2576","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0802.2576","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"arxiv_version","alias_value":"0802.2576v2","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0802.2576","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"pith_short_12","alias_value":"NC7GTMGJCBJ3","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"NC7GTMGJCBJ3C4CB","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"NC7GTMGJ","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:NC7GTMGJCBJ3C4CB4QOGR3YP7M","target":"record","payload":{"canonical_record":{"source":{"id":"0802.2576","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-02-19T02:04:49Z","cross_cats_sorted":[],"title_canon_sha256":"e79c77e598a1cba77fd4e920f773d4d322661bb8a8fa686c706e361c3214f7df","abstract_canon_sha256":"7a31f4835c66b4db52f72cb09135a4595ba453b27232d9bdce01c930be0eef43"},"schema_version":"1.0"},"canonical_sha256":"68be69b0c91053b17041e41c68ef0ffb3d900e66d1e8db5c8b8c92a9a6300b22","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:49.446689Z","signature_b64":"RsEuLf+AmkBH2//eNI7zmQtgm+r/q07RK/cbqcQ6/Wxmlyr7FsJ0+A5mjZFlq2X+mW6NfXA78g+YcZrEKcWVAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68be69b0c91053b17041e41c68ef0ffb3d900e66d1e8db5c8b8c92a9a6300b22","last_reissued_at":"2026-05-18T04:11:49.446192Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:49.446192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0802.2576","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:11:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"22+JlEljUOh5JySWO63uifZdWTKKkTm7xD+cAdItyFFK5zOeR5LZ+gdjzgcxQzQs5sbv00EphXQ8pcf/ae/KDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:08:16.201296Z"},"content_sha256":"ea47e4d0038170a48255e3a9f7d1c5e6575af13a92c100e59297b532d44f76c5","schema_version":"1.0","event_id":"sha256:ea47e4d0038170a48255e3a9f7d1c5e6575af13a92c100e59297b532d44f76c5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:NC7GTMGJCBJ3C4CB4QOGR3YP7M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Simplicial matrix-tree theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Art M. Duval, Caroline J. Klivans, Jeremy L. Martin","submitted_at":"2008-02-19T02:04:49Z","abstract_excerpt":"We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\\Delta$ and replaci"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.2576","kind":"arxiv","version":2},"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":"2026-05-18T04:11:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZdgfltrVo6EZgzQR+wDWNiejBi7KvBgxQgtvtll0Rpvhw9MqHh7vK83W3y8iOk84er81HaqUH+akmxr9aKb9Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:08:16.201956Z"},"content_sha256":"0cf28043631eae888cc600823fa6c21f496129f1bac65739b8aaedb27162992e","schema_version":"1.0","event_id":"sha256:0cf28043631eae888cc600823fa6c21f496129f1bac65739b8aaedb27162992e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NC7GTMGJCBJ3C4CB4QOGR3YP7M/bundle.json","state_url":"https://pith.science/pith/NC7GTMGJCBJ3C4CB4QOGR3YP7M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NC7GTMGJCBJ3C4CB4QOGR3YP7M/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":"2026-06-05T09:08:16Z","links":{"resolver":"https://pith.science/pith/NC7GTMGJCBJ3C4CB4QOGR3YP7M","bundle":"https://pith.science/pith/NC7GTMGJCBJ3C4CB4QOGR3YP7M/bundle.json","state":"https://pith.science/pith/NC7GTMGJCBJ3C4CB4QOGR3YP7M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NC7GTMGJCBJ3C4CB4QOGR3YP7M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:NC7GTMGJCBJ3C4CB4QOGR3YP7M","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":"7a31f4835c66b4db52f72cb09135a4595ba453b27232d9bdce01c930be0eef43","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-02-19T02:04:49Z","title_canon_sha256":"e79c77e598a1cba77fd4e920f773d4d322661bb8a8fa686c706e361c3214f7df"},"schema_version":"1.0","source":{"id":"0802.2576","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0802.2576","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"arxiv_version","alias_value":"0802.2576v2","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0802.2576","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"pith_short_12","alias_value":"NC7GTMGJCBJ3","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"NC7GTMGJCBJ3C4CB","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"NC7GTMGJ","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:0cf28043631eae888cc600823fa6c21f496129f1bac65739b8aaedb27162992e","target":"graph","created_at":"2026-05-18T04:11:49Z","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 generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\\Delta$ and replaci","authors_text":"Art M. Duval, Caroline J. Klivans, Jeremy L. Martin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-02-19T02:04:49Z","title":"Simplicial matrix-tree theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.2576","kind":"arxiv","version":2},"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:ea47e4d0038170a48255e3a9f7d1c5e6575af13a92c100e59297b532d44f76c5","target":"record","created_at":"2026-05-18T04:11:49Z","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":"7a31f4835c66b4db52f72cb09135a4595ba453b27232d9bdce01c930be0eef43","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-02-19T02:04:49Z","title_canon_sha256":"e79c77e598a1cba77fd4e920f773d4d322661bb8a8fa686c706e361c3214f7df"},"schema_version":"1.0","source":{"id":"0802.2576","kind":"arxiv","version":2}},"canonical_sha256":"68be69b0c91053b17041e41c68ef0ffb3d900e66d1e8db5c8b8c92a9a6300b22","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68be69b0c91053b17041e41c68ef0ffb3d900e66d1e8db5c8b8c92a9a6300b22","first_computed_at":"2026-05-18T04:11:49.446192Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:49.446192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RsEuLf+AmkBH2//eNI7zmQtgm+r/q07RK/cbqcQ6/Wxmlyr7FsJ0+A5mjZFlq2X+mW6NfXA78g+YcZrEKcWVAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:49.446689Z","signed_message":"canonical_sha256_bytes"},"source_id":"0802.2576","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea47e4d0038170a48255e3a9f7d1c5e6575af13a92c100e59297b532d44f76c5","sha256:0cf28043631eae888cc600823fa6c21f496129f1bac65739b8aaedb27162992e"],"state_sha256":"a9ad522a6f1d50f8b72397cccc2b3b66c3b94ed97205f6f2c419373b8bab6ebb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"knk+OFsaanNAiF/hvJke6ivijxT3cUOv+uvg412TqKuE44eGG0Ek56zsRmld/GJ6DidijuS8fIzL8n1jLX2oBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T09:08:16.205050Z","bundle_sha256":"1a3fc955dab391bef7b6cdbeded6ca3f8867a7ce0d84b7654212d8117034c475"}}