{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XIFUVVMRK5K4SFDEJTQXARZHGG","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":"06cc13e4601ad61e53c7dcb51f7a903c3cf863901c65eb4c4bee10f8d823e5f2","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2013-07-08T15:59:00Z","title_canon_sha256":"dae110c2f7ad76886a24d0a2d25021adddf8502d01af5b1aa5700bd759ce1ed5"},"schema_version":"1.0","source":{"id":"1307.2137","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.2137","created_at":"2026-05-18T01:16:37Z"},{"alias_kind":"arxiv_version","alias_value":"1307.2137v4","created_at":"2026-05-18T01:16:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.2137","created_at":"2026-05-18T01:16:37Z"},{"alias_kind":"pith_short_12","alias_value":"XIFUVVMRK5K4","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XIFUVVMRK5K4SFDE","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XIFUVVMR","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:f210141430ee91e533e3f45b28cfad6cb2fa4d7182f4a4dbf3929587de6a49a6","target":"graph","created_at":"2026-05-18T01:16:37Z","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":"This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil.","authors_text":"I.P. Goulden, Jonathan Novak, Mathieu Guay-Paquet","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2013-07-08T15:59:00Z","title":"Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2137","kind":"arxiv","version":4},"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":"sha256:8dc180a16b570442a3edf94690bc93cc4515363e909c1389ec4b4037da65f8ee","target":"record","created_at":"2026-05-18T01:16:37Z","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":"06cc13e4601ad61e53c7dcb51f7a903c3cf863901c65eb4c4bee10f8d823e5f2","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2013-07-08T15:59:00Z","title_canon_sha256":"dae110c2f7ad76886a24d0a2d25021adddf8502d01af5b1aa5700bd759ce1ed5"},"schema_version":"1.0","source":{"id":"1307.2137","kind":"arxiv","version":4}},"canonical_sha256":"ba0b4ad5915755c914644ce1704727319f22de8461ff8d969c8756a3694ef3b9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ba0b4ad5915755c914644ce1704727319f22de8461ff8d969c8756a3694ef3b9","first_computed_at":"2026-05-18T01:16:37.771670Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:37.771670Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ly2XjhuJ9Y1+wPtpf9jqw+mVVW8tVT/noPEIb/xtwRi0ITt/nCUTHU7rou8YW1iIJt7vDko+m6y1ogHsFiOJBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:37.772307Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.2137","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8dc180a16b570442a3edf94690bc93cc4515363e909c1389ec4b4037da65f8ee","sha256:f210141430ee91e533e3f45b28cfad6cb2fa4d7182f4a4dbf3929587de6a49a6"],"state_sha256":"1d4016a6608c4cb6393bca02470b40181825c25fdaa72efc33cd708e6d1250d8"}