{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:CBSESLSYKMLOEPQEXVSKLRJ3WK","short_pith_number":"pith:CBSESLSY","canonical_record":{"source":{"id":"1612.02884","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-09T01:27:55Z","cross_cats_sorted":[],"title_canon_sha256":"518f53e45a4b430bb7178b86fa3e085daca5ce552e0d737b6508d510e6378bce","abstract_canon_sha256":"a5d3dc6c61bc28e943b66ecce78983d1732eaeebeb5e94106b8d697a0e6b0d7f"},"schema_version":"1.0"},"canonical_sha256":"1064492e585316e23e04bd64a5c53bb2bb816315dee3eeed120eb762d2b1554b","source":{"kind":"arxiv","id":"1612.02884","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.02884","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"arxiv_version","alias_value":"1612.02884v1","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02884","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"pith_short_12","alias_value":"CBSESLSYKMLO","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CBSESLSYKMLOEPQE","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CBSESLSY","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:CBSESLSYKMLOEPQEXVSKLRJ3WK","target":"record","payload":{"canonical_record":{"source":{"id":"1612.02884","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-09T01:27:55Z","cross_cats_sorted":[],"title_canon_sha256":"518f53e45a4b430bb7178b86fa3e085daca5ce552e0d737b6508d510e6378bce","abstract_canon_sha256":"a5d3dc6c61bc28e943b66ecce78983d1732eaeebeb5e94106b8d697a0e6b0d7f"},"schema_version":"1.0"},"canonical_sha256":"1064492e585316e23e04bd64a5c53bb2bb816315dee3eeed120eb762d2b1554b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:28.064666Z","signature_b64":"x/lxA1F+TrU0JMzRcX6giK9RUaeLPKQftQODVElYLX4POu+V6ra1XOUoVwH1w611PJ8b1rgGhrxFpUA3Jcv/Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1064492e585316e23e04bd64a5c53bb2bb816315dee3eeed120eb762d2b1554b","last_reissued_at":"2026-05-18T00:55:28.064153Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:28.064153Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.02884","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":"2026-05-18T00:55:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bsKmtQgHkHwNSpPO532hyss7HHYHr+9gX4GFItHjF88NprhjiQAc5a3CY5Hj5lYKDsgh21btWf/3zaBEWa7gBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:05:43.812479Z"},"content_sha256":"beba2d8d63f79bec8d4b6bc0157ae5c1bae7c0d7c502e32beb700882e072f442","schema_version":"1.0","event_id":"sha256:beba2d8d63f79bec8d4b6bc0157ae5c1bae7c0d7c502e32beb700882e072f442"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:CBSESLSYKMLOEPQEXVSKLRJ3WK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"W-Operator and Differential Equation for 3-Hurwitz Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hao Sun","submitted_at":"2016-12-09T01:27:55Z","abstract_excerpt":"We consider a new type of Hurwitz number, the number of ordered transitive factorizations of an arbitrary permutation into d-cycles. In this paper, we focus on the special case d = 3. The minimal number of transitive factorizations of any permutation into 3-cycles has been worked out by David, Goulden and Jackson. Also, such factorizations for transpositions, the case d = 2, have been considered by Crescimanno and Taylor. Goulden and Jackson have proved the differential equation for the generating series of simple Hurwitz numbers. Based on their results, we use W-operator to prove a differenti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02884","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":"2026-05-18T00:55:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YPpQGAPqdRfoL8XKqn2hbxySY0yqmWf1LhsZjnE4Xtba/isZ5fXTWBEm0r2Nz9Icqi836/X1Rlf/kw/kuVHaCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:05:43.813011Z"},"content_sha256":"cffbd996995e1b6ed83e53d5a076d3c231597786a83b038b21f802829394a0ba","schema_version":"1.0","event_id":"sha256:cffbd996995e1b6ed83e53d5a076d3c231597786a83b038b21f802829394a0ba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CBSESLSYKMLOEPQEXVSKLRJ3WK/bundle.json","state_url":"https://pith.science/pith/CBSESLSYKMLOEPQEXVSKLRJ3WK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CBSESLSYKMLOEPQEXVSKLRJ3WK/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":"2026-05-26T16:05:43Z","links":{"resolver":"https://pith.science/pith/CBSESLSYKMLOEPQEXVSKLRJ3WK","bundle":"https://pith.science/pith/CBSESLSYKMLOEPQEXVSKLRJ3WK/bundle.json","state":"https://pith.science/pith/CBSESLSYKMLOEPQEXVSKLRJ3WK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CBSESLSYKMLOEPQEXVSKLRJ3WK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CBSESLSYKMLOEPQEXVSKLRJ3WK","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":"a5d3dc6c61bc28e943b66ecce78983d1732eaeebeb5e94106b8d697a0e6b0d7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-09T01:27:55Z","title_canon_sha256":"518f53e45a4b430bb7178b86fa3e085daca5ce552e0d737b6508d510e6378bce"},"schema_version":"1.0","source":{"id":"1612.02884","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.02884","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"arxiv_version","alias_value":"1612.02884v1","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02884","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"pith_short_12","alias_value":"CBSESLSYKMLO","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CBSESLSYKMLOEPQE","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CBSESLSY","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:cffbd996995e1b6ed83e53d5a076d3c231597786a83b038b21f802829394a0ba","target":"graph","created_at":"2026-05-18T00:55:28Z","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 consider a new type of Hurwitz number, the number of ordered transitive factorizations of an arbitrary permutation into d-cycles. In this paper, we focus on the special case d = 3. The minimal number of transitive factorizations of any permutation into 3-cycles has been worked out by David, Goulden and Jackson. Also, such factorizations for transpositions, the case d = 2, have been considered by Crescimanno and Taylor. Goulden and Jackson have proved the differential equation for the generating series of simple Hurwitz numbers. Based on their results, we use W-operator to prove a differenti","authors_text":"Hao Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-09T01:27:55Z","title":"W-Operator and Differential Equation for 3-Hurwitz Number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02884","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:beba2d8d63f79bec8d4b6bc0157ae5c1bae7c0d7c502e32beb700882e072f442","target":"record","created_at":"2026-05-18T00:55:28Z","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":"a5d3dc6c61bc28e943b66ecce78983d1732eaeebeb5e94106b8d697a0e6b0d7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-09T01:27:55Z","title_canon_sha256":"518f53e45a4b430bb7178b86fa3e085daca5ce552e0d737b6508d510e6378bce"},"schema_version":"1.0","source":{"id":"1612.02884","kind":"arxiv","version":1}},"canonical_sha256":"1064492e585316e23e04bd64a5c53bb2bb816315dee3eeed120eb762d2b1554b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1064492e585316e23e04bd64a5c53bb2bb816315dee3eeed120eb762d2b1554b","first_computed_at":"2026-05-18T00:55:28.064153Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:28.064153Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x/lxA1F+TrU0JMzRcX6giK9RUaeLPKQftQODVElYLX4POu+V6ra1XOUoVwH1w611PJ8b1rgGhrxFpUA3Jcv/Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:28.064666Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.02884","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:beba2d8d63f79bec8d4b6bc0157ae5c1bae7c0d7c502e32beb700882e072f442","sha256:cffbd996995e1b6ed83e53d5a076d3c231597786a83b038b21f802829394a0ba"],"state_sha256":"91a9e864c1c77e84eb002671fd96c9c14e8ca1e5208da1692483faed3bbf2381"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4YpPoZIi+RtsfbWSHoBAWhj6sfmY/3jEDuT+qeH0XiaoUi70lM25vuokRwpSVvHpDtFvnuIpud5EyEV4UbSbCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T16:05:43.815590Z","bundle_sha256":"3fb4b0d2a5d93746c821c20e10c0c298a0c7ee430301710254662adc8801325b"}}