{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:JYUBGFSXFXOW2EHDU2NFTURUCI","short_pith_number":"pith:JYUBGFSX","canonical_record":{"source":{"id":"1611.04700","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-15T04:10:34Z","cross_cats_sorted":[],"title_canon_sha256":"c54064e1f4cd4f8f6faee68a7ddfcebd4c4f91fc71985ecf974280e65cdd6cf0","abstract_canon_sha256":"f55caac7decc89ee0db044250bd0fba6d4747d8c7bfdc0130c5699de964783d6"},"schema_version":"1.0"},"canonical_sha256":"4e281316572ddd6d10e3a69a59d2341201d0e5043ef3b3cdb13a44017a3a1e69","source":{"kind":"arxiv","id":"1611.04700","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04700","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04700v2","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04700","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"pith_short_12","alias_value":"JYUBGFSXFXOW","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JYUBGFSXFXOW2EHD","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JYUBGFSX","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:JYUBGFSXFXOW2EHDU2NFTURUCI","target":"record","payload":{"canonical_record":{"source":{"id":"1611.04700","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-15T04:10:34Z","cross_cats_sorted":[],"title_canon_sha256":"c54064e1f4cd4f8f6faee68a7ddfcebd4c4f91fc71985ecf974280e65cdd6cf0","abstract_canon_sha256":"f55caac7decc89ee0db044250bd0fba6d4747d8c7bfdc0130c5699de964783d6"},"schema_version":"1.0"},"canonical_sha256":"4e281316572ddd6d10e3a69a59d2341201d0e5043ef3b3cdb13a44017a3a1e69","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:59.253462Z","signature_b64":"6Rc9HmGaysqfOhlgl8SG0mO/hGuPFaga46pNorp7KZTkmD0hp0OhGCydzBvwmsUFmd77rt5MdHRdRr07kirQAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e281316572ddd6d10e3a69a59d2341201d0e5043ef3b3cdb13a44017a3a1e69","last_reissued_at":"2026-05-17T23:42:59.253017Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:59.253017Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.04700","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":"2026-05-17T23:42:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z5sgghl8FSbzAyDyPem7DnXOrHUltTJQDhfsfZVDhxJrAr/PA0pNzmT8hIdfUNHg2Lj2LL6BMxM5NOC+duf3Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:57:24.950826Z"},"content_sha256":"706be32528d3c736f9a370d73d955ed0ce9984e24f32b00d741f02239f693aa8","schema_version":"1.0","event_id":"sha256:706be32528d3c736f9a370d73d955ed0ce9984e24f32b00d741f02239f693aa8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:JYUBGFSXFXOW2EHDU2NFTURUCI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Formula about W-operator and Its Application to Hurwitz Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hao Sun","submitted_at":"2016-11-15T04:10:34Z","abstract_excerpt":"W-operators are differential operators on the polynomial ring. Mironov, Morosov and Natanzon construct the generalized Hurwitz numbers. They use the W-operator to prove a formula for the generating function of the generalized Hurwitz numbers. A special example of the W-operator is the cut-and-join operator. Goulden and Jackson use the cut-and-join operator to calculate the simple Hurwitz number. In this paper, we study the relation between W-operator W([d]) and the central elements in Sn. Based on the relation we find, we give another proof about a differential equation of the generating funct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04700","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":"2026-05-17T23:42:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Eol5KO6hT41VGZssyjK2bpiyWSjJI3YiFrlj3MGqI6wFkKguwv8X+B0BB+H+0WuKig+p1i4c5t4j85mMjcP0Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:57:24.951254Z"},"content_sha256":"9a5f926cc82e1e132b480491a9fc503a6cf3c4b39cc40e7231fc8eab3aa0fd81","schema_version":"1.0","event_id":"sha256:9a5f926cc82e1e132b480491a9fc503a6cf3c4b39cc40e7231fc8eab3aa0fd81"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JYUBGFSXFXOW2EHDU2NFTURUCI/bundle.json","state_url":"https://pith.science/pith/JYUBGFSXFXOW2EHDU2NFTURUCI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JYUBGFSXFXOW2EHDU2NFTURUCI/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":"2026-05-26T18:57:24Z","links":{"resolver":"https://pith.science/pith/JYUBGFSXFXOW2EHDU2NFTURUCI","bundle":"https://pith.science/pith/JYUBGFSXFXOW2EHDU2NFTURUCI/bundle.json","state":"https://pith.science/pith/JYUBGFSXFXOW2EHDU2NFTURUCI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JYUBGFSXFXOW2EHDU2NFTURUCI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:JYUBGFSXFXOW2EHDU2NFTURUCI","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":"f55caac7decc89ee0db044250bd0fba6d4747d8c7bfdc0130c5699de964783d6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-15T04:10:34Z","title_canon_sha256":"c54064e1f4cd4f8f6faee68a7ddfcebd4c4f91fc71985ecf974280e65cdd6cf0"},"schema_version":"1.0","source":{"id":"1611.04700","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04700","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04700v2","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04700","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"pith_short_12","alias_value":"JYUBGFSXFXOW","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JYUBGFSXFXOW2EHD","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JYUBGFSX","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:9a5f926cc82e1e132b480491a9fc503a6cf3c4b39cc40e7231fc8eab3aa0fd81","target":"graph","created_at":"2026-05-17T23:42:59Z","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":"W-operators are differential operators on the polynomial ring. Mironov, Morosov and Natanzon construct the generalized Hurwitz numbers. They use the W-operator to prove a formula for the generating function of the generalized Hurwitz numbers. A special example of the W-operator is the cut-and-join operator. Goulden and Jackson use the cut-and-join operator to calculate the simple Hurwitz number. In this paper, we study the relation between W-operator W([d]) and the central elements in Sn. Based on the relation we find, we give another proof about a differential equation of the generating funct","authors_text":"Hao Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-15T04:10:34Z","title":"A Formula about W-operator and Its Application to Hurwitz Number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04700","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":"sha256:706be32528d3c736f9a370d73d955ed0ce9984e24f32b00d741f02239f693aa8","target":"record","created_at":"2026-05-17T23:42:59Z","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":"f55caac7decc89ee0db044250bd0fba6d4747d8c7bfdc0130c5699de964783d6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-15T04:10:34Z","title_canon_sha256":"c54064e1f4cd4f8f6faee68a7ddfcebd4c4f91fc71985ecf974280e65cdd6cf0"},"schema_version":"1.0","source":{"id":"1611.04700","kind":"arxiv","version":2}},"canonical_sha256":"4e281316572ddd6d10e3a69a59d2341201d0e5043ef3b3cdb13a44017a3a1e69","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e281316572ddd6d10e3a69a59d2341201d0e5043ef3b3cdb13a44017a3a1e69","first_computed_at":"2026-05-17T23:42:59.253017Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:59.253017Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6Rc9HmGaysqfOhlgl8SG0mO/hGuPFaga46pNorp7KZTkmD0hp0OhGCydzBvwmsUFmd77rt5MdHRdRr07kirQAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:59.253462Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.04700","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:706be32528d3c736f9a370d73d955ed0ce9984e24f32b00d741f02239f693aa8","sha256:9a5f926cc82e1e132b480491a9fc503a6cf3c4b39cc40e7231fc8eab3aa0fd81"],"state_sha256":"660199a2639575dcbce8f8c0ac3d07d44fd15ade42a915d04c59106ad6be77be"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aJxzNKs+mSXttcx8bJvJFQCjBg30lG65C2wS9rV8L54Z0Ju0U8MQGe/FG2JClSztPArWw8L4x+68RMZSkzHXCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T18:57:24.953983Z","bundle_sha256":"5933ba5cf8148915d0f556cc9519de3872a3cde370c93166f2db7dc634a732a5"}}