{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:JLDXLYSVSP5Z3Z6V4JVYGMIBRM","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":"7fe7d8a815d361116f441beb51d5bada688caeeabf9e777cba2246c1012b9072","cross_cats_sorted":["math.CO","math.MP","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-03-06T10:31:23Z","title_canon_sha256":"f1cddc5816c3aae26df1249f70ea9c61b1d43d5acb2c93981921f365a0e4e869"},"schema_version":"1.0","source":{"id":"1603.01815","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.01815","created_at":"2026-05-18T01:19:33Z"},{"alias_kind":"arxiv_version","alias_value":"1603.01815v1","created_at":"2026-05-18T01:19:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.01815","created_at":"2026-05-18T01:19:33Z"},{"alias_kind":"pith_short_12","alias_value":"JLDXLYSVSP5Z","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JLDXLYSVSP5Z3Z6V","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JLDXLYSV","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:ef06e7f90de89a3836e1cd3b99cc0680331b7dc431fdd12f8bc2c9b14abfc78d","target":"graph","created_at":"2026-05-18T01:19:33Z","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 study two different one-parameter generalizations of Littlewood--Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions are closely related to puzzles, originally introduced by Knutson and Tao in their work on the equivariant cohomology of the Grassmannian.","authors_text":"Michael Wheeler, Paul Zinn-Justin","cross_cats":["math.CO","math.MP","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-03-06T10:31:23Z","title":"Hall polynomials, inverse Kostka polynomials and puzzles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01815","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":"sha256:71c28ddc11c4189b716fcc66deb2ed11b55cf61a7d6858e088ba3173503a9387","target":"record","created_at":"2026-05-18T01:19:33Z","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":"7fe7d8a815d361116f441beb51d5bada688caeeabf9e777cba2246c1012b9072","cross_cats_sorted":["math.CO","math.MP","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-03-06T10:31:23Z","title_canon_sha256":"f1cddc5816c3aae26df1249f70ea9c61b1d43d5acb2c93981921f365a0e4e869"},"schema_version":"1.0","source":{"id":"1603.01815","kind":"arxiv","version":1}},"canonical_sha256":"4ac775e25593fb9de7d5e26b8331018b161596011e083a458517f19d4a991a64","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ac775e25593fb9de7d5e26b8331018b161596011e083a458517f19d4a991a64","first_computed_at":"2026-05-18T01:19:33.591276Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:33.591276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"emeAZz48QDFg2eevH+xVTLCeR23f3vyqlAfpq0hlI9HBFd7/XizImq5IcShzKSvJWsVtNKdgIcGDRr9nBy7kCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:33.591709Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.01815","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71c28ddc11c4189b716fcc66deb2ed11b55cf61a7d6858e088ba3173503a9387","sha256:ef06e7f90de89a3836e1cd3b99cc0680331b7dc431fdd12f8bc2c9b14abfc78d"],"state_sha256":"79865acef5a011ee2cef73d99569546d343ee7c266fd4b8857eb966fc4bd317d"}