{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:SAJCBIKLIRUSLKSINUMQRRKASB","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":"9b3d8c0f3e470429e9d2f363bf163ec60756aed665b17562ab1a563d9db1e7aa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-07T21:36:38Z","title_canon_sha256":"1765a911327f269248163eb1b9022ba879e961fa5fd27cce59ac9a6fe8b8ad51"},"schema_version":"1.0","source":{"id":"1809.02685","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.02685","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"arxiv_version","alias_value":"1809.02685v1","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.02685","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"pith_short_12","alias_value":"SAJCBIKLIRUS","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"SAJCBIKLIRUSLKSI","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"SAJCBIKL","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:639b05b30f6a8c2dfe46f36c9e08a3c516daee2449e36b7fe2255ca5af16b2fa","target":"graph","created_at":"2026-05-18T00:06:15Z","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":"Refined versions, analytic and combinatorial, are given for classical integer partition theorems. The examples include the Rogers-Ramanujan identities, the Gollnitz-Gordon identities, Euler's odd=distinct theorem, and the Andrews-Gordon identities. Generalizations of each of these theorems are given where a single part is \"marked\" or weighted. This allows a single part to be replaced by a new larger part, \"shifting\" a part, and analogous combinatorial results are given in each case. Versions are also given for marking a sum of parts.","authors_text":"Dennis Stanton, Kathleen O'Hara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-07T21:36:38Z","title":"Marking and shifting a part in partition theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02685","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":"sha256:7d66f6d747a572a499b62015183075524427848c775cd61cde4bf7c68c5a59f2","target":"record","created_at":"2026-05-18T00:06:15Z","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":"9b3d8c0f3e470429e9d2f363bf163ec60756aed665b17562ab1a563d9db1e7aa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-07T21:36:38Z","title_canon_sha256":"1765a911327f269248163eb1b9022ba879e961fa5fd27cce59ac9a6fe8b8ad51"},"schema_version":"1.0","source":{"id":"1809.02685","kind":"arxiv","version":1}},"canonical_sha256":"901220a14b446925aa486d1908c54090509dd529212af7b1c4c283386b0cb513","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"901220a14b446925aa486d1908c54090509dd529212af7b1c4c283386b0cb513","first_computed_at":"2026-05-18T00:06:15.611600Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:15.611600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GqlwtHW4d+gY1v5RbUNoy2zsrILNzuKQxy0gpwujVO7TFQwrVZh0LsHIi2MXcq+Yu6VhJVAeBq9w8iKqaiDJCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:15.612074Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.02685","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7d66f6d747a572a499b62015183075524427848c775cd61cde4bf7c68c5a59f2","sha256:639b05b30f6a8c2dfe46f36c9e08a3c516daee2449e36b7fe2255ca5af16b2fa"],"state_sha256":"9dc1957d3c1653aa398fb9daa01800d106276260df97ffb23055d4d2db254cc6"}