{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:Y5VQVN2T2E5AX6YMTBOI2OSACT","short_pith_number":"pith:Y5VQVN2T","canonical_record":{"source":{"id":"1006.5172","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-26T22:42:28Z","cross_cats_sorted":[],"title_canon_sha256":"d9677f98d2e2eb5ced92e99a293881663becf77c03df8d691ca71eccf6312175","abstract_canon_sha256":"fbc3e90c78bd4f1192aa0417c518c9c1e8a8c6cc35736eb662993999718f3587"},"schema_version":"1.0"},"canonical_sha256":"c76b0ab753d13a0bfb0c985c8d3a4014e28a0e1533cc546f0188bcce7467c66a","source":{"kind":"arxiv","id":"1006.5172","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.5172","created_at":"2026-05-18T03:57:37Z"},{"alias_kind":"arxiv_version","alias_value":"1006.5172v1","created_at":"2026-05-18T03:57:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.5172","created_at":"2026-05-18T03:57:37Z"},{"alias_kind":"pith_short_12","alias_value":"Y5VQVN2T2E5A","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"Y5VQVN2T2E5AX6YM","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"Y5VQVN2T","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:Y5VQVN2T2E5AX6YMTBOI2OSACT","target":"record","payload":{"canonical_record":{"source":{"id":"1006.5172","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-26T22:42:28Z","cross_cats_sorted":[],"title_canon_sha256":"d9677f98d2e2eb5ced92e99a293881663becf77c03df8d691ca71eccf6312175","abstract_canon_sha256":"fbc3e90c78bd4f1192aa0417c518c9c1e8a8c6cc35736eb662993999718f3587"},"schema_version":"1.0"},"canonical_sha256":"c76b0ab753d13a0bfb0c985c8d3a4014e28a0e1533cc546f0188bcce7467c66a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:37.399433Z","signature_b64":"4dFT1jK4ITianbhaMc1fOXQNjls4g+t3P6SHYuvqhcWZ9BnZh5SiuMoIw/f6zU2dwtcBFa8ScCuNqhlu4QzgAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c76b0ab753d13a0bfb0c985c8d3a4014e28a0e1533cc546f0188bcce7467c66a","last_reissued_at":"2026-05-18T03:57:37.398739Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:37.398739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1006.5172","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:57:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Weru4cz7+6gmg6VFgSsub3EwrWwL9WNrSU03MJK9U7960oGeChTgnZMn2w1EMA18JJG038EYAg4bCuqdC7SaDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T17:07:21.432275Z"},"content_sha256":"5a954287bbb8118ceea654e65bf01e82dd61c75579338fdd18442ad76e2e9b57","schema_version":"1.0","event_id":"sha256:5a954287bbb8118ceea654e65bf01e82dd61c75579338fdd18442ad76e2e9b57"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:Y5VQVN2T2E5AX6YMTBOI2OSACT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting unicellular maps on non-orientable surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guillaume Chapuy, Olivier Bernardi","submitted_at":"2010-06-26T22:42:28Z","abstract_excerpt":"A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable unicellular maps of fixed topology. In particular, we give exact formulas for the precubic case (all vertices of degree 1 or 3), and asymptotic formulas for the general case, when"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5172","kind":"arxiv","version":1},"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":"2026-05-18T03:57:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1nhDj0zJDkEQmEpt7N6KWi7N9X4pZ5Ac3TN8UObrhb7H8YiT5+FA4DIdzYUkKk2cUwAzU3opV2BuieWml0RUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T17:07:21.432905Z"},"content_sha256":"86f1a41867bfbf8487e9696a525354ec1cdfb93efe55031b9ce980fe9b46316b","schema_version":"1.0","event_id":"sha256:86f1a41867bfbf8487e9696a525354ec1cdfb93efe55031b9ce980fe9b46316b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y5VQVN2T2E5AX6YMTBOI2OSACT/bundle.json","state_url":"https://pith.science/pith/Y5VQVN2T2E5AX6YMTBOI2OSACT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y5VQVN2T2E5AX6YMTBOI2OSACT/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":"2026-06-09T17:07:21Z","links":{"resolver":"https://pith.science/pith/Y5VQVN2T2E5AX6YMTBOI2OSACT","bundle":"https://pith.science/pith/Y5VQVN2T2E5AX6YMTBOI2OSACT/bundle.json","state":"https://pith.science/pith/Y5VQVN2T2E5AX6YMTBOI2OSACT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y5VQVN2T2E5AX6YMTBOI2OSACT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:Y5VQVN2T2E5AX6YMTBOI2OSACT","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":"fbc3e90c78bd4f1192aa0417c518c9c1e8a8c6cc35736eb662993999718f3587","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-26T22:42:28Z","title_canon_sha256":"d9677f98d2e2eb5ced92e99a293881663becf77c03df8d691ca71eccf6312175"},"schema_version":"1.0","source":{"id":"1006.5172","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.5172","created_at":"2026-05-18T03:57:37Z"},{"alias_kind":"arxiv_version","alias_value":"1006.5172v1","created_at":"2026-05-18T03:57:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.5172","created_at":"2026-05-18T03:57:37Z"},{"alias_kind":"pith_short_12","alias_value":"Y5VQVN2T2E5A","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"Y5VQVN2T2E5AX6YM","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"Y5VQVN2T","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:86f1a41867bfbf8487e9696a525354ec1cdfb93efe55031b9ce980fe9b46316b","target":"graph","created_at":"2026-05-18T03:57:37Z","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":"A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable unicellular maps of fixed topology. In particular, we give exact formulas for the precubic case (all vertices of degree 1 or 3), and asymptotic formulas for the general case, when","authors_text":"Guillaume Chapuy, Olivier Bernardi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-26T22:42:28Z","title":"Counting unicellular maps on non-orientable surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5172","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":"sha256:5a954287bbb8118ceea654e65bf01e82dd61c75579338fdd18442ad76e2e9b57","target":"record","created_at":"2026-05-18T03:57:37Z","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":"fbc3e90c78bd4f1192aa0417c518c9c1e8a8c6cc35736eb662993999718f3587","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-26T22:42:28Z","title_canon_sha256":"d9677f98d2e2eb5ced92e99a293881663becf77c03df8d691ca71eccf6312175"},"schema_version":"1.0","source":{"id":"1006.5172","kind":"arxiv","version":1}},"canonical_sha256":"c76b0ab753d13a0bfb0c985c8d3a4014e28a0e1533cc546f0188bcce7467c66a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c76b0ab753d13a0bfb0c985c8d3a4014e28a0e1533cc546f0188bcce7467c66a","first_computed_at":"2026-05-18T03:57:37.398739Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:37.398739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4dFT1jK4ITianbhaMc1fOXQNjls4g+t3P6SHYuvqhcWZ9BnZh5SiuMoIw/f6zU2dwtcBFa8ScCuNqhlu4QzgAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:37.399433Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.5172","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a954287bbb8118ceea654e65bf01e82dd61c75579338fdd18442ad76e2e9b57","sha256:86f1a41867bfbf8487e9696a525354ec1cdfb93efe55031b9ce980fe9b46316b"],"state_sha256":"5e481a0997dd04bf24b3363e5431823433a9164d353d9d58b33e192d7b393175"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rVCRICvsne78td4VOiXfw/hKb/Myb/fw97zXUI40fRTRIQFn4uKi2rktgJ1YAmbviCchpZZWgtjhn2ZhGdTnAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T17:07:21.435246Z","bundle_sha256":"af4e0cf293a18154e1051cf22cf6668fe6252f0d609a27b978202caf87dfd114"}}