{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:VVC72XQ6FOE4P64C5XSITOEIIT","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":"43368ca1785305fbd4a9cb8b58275a71656db7d3666bb4f42a8fc4f218ca64f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2005-08-11T01:20:56Z","title_canon_sha256":"d7090ffc6210b73e304c33233081b56528672cbab52332261518ae34d7485e7a"},"schema_version":"1.0","source":{"id":"math/0508192","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0508192","created_at":"2026-05-18T04:33:27Z"},{"alias_kind":"arxiv_version","alias_value":"math/0508192v2","created_at":"2026-05-18T04:33:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0508192","created_at":"2026-05-18T04:33:27Z"},{"alias_kind":"pith_short_12","alias_value":"VVC72XQ6FOE4","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"VVC72XQ6FOE4P64C","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"VVC72XQ6","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:b7a00ba04b5749e18174573fcc57d603cdca20fb3191e09aef625842253d2c48","target":"graph","created_at":"2026-05-18T04:33:27Z","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":"In this paper, we study Grothendieck polynomials from a combinatorial viewpoint. We introduce the factorial Grothendieck polynomials, analogues of the factorial Schur functions and present some of their properties, and use them to produce a generalisation of a Littlewood-Richardson rule for Grothendieck polynomials.","authors_text":"Peter J. McNamara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2005-08-11T01:20:56Z","title":"Factorial Grothendieck Polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508192","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:d944c17bf245dd1a708229060e588409ab412cb4e0b1084d082e5587f98804ed","target":"record","created_at":"2026-05-18T04:33:27Z","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":"43368ca1785305fbd4a9cb8b58275a71656db7d3666bb4f42a8fc4f218ca64f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2005-08-11T01:20:56Z","title_canon_sha256":"d7090ffc6210b73e304c33233081b56528672cbab52332261518ae34d7485e7a"},"schema_version":"1.0","source":{"id":"math/0508192","kind":"arxiv","version":2}},"canonical_sha256":"ad45fd5e1e2b89c7fb82ede489b88844c27c7f10083b7429501809c225fea3c5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad45fd5e1e2b89c7fb82ede489b88844c27c7f10083b7429501809c225fea3c5","first_computed_at":"2026-05-18T04:33:27.526136Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:33:27.526136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1B0tpTp7K1qrmySxC1h1uXxwwh/76ezIV6Q5jrsFE49TU7AHxJhju3urWObzSb29sLwm9W4YwISfXYTPQkW4DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:33:27.526741Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0508192","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d944c17bf245dd1a708229060e588409ab412cb4e0b1084d082e5587f98804ed","sha256:b7a00ba04b5749e18174573fcc57d603cdca20fb3191e09aef625842253d2c48"],"state_sha256":"8a7c2b9f4270d2aa7268c9528fba36750d86ff00d6a56c896886bcbf39b09422"}