{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2J7LZXUKKZOD74QH5OUXBRUQ72","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":"d8115f4fa0956621bbf30c70a377c2136a83e502920531a00f9d9a9e73906978","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-28T08:53:22Z","title_canon_sha256":"1f646e6c9870b8224cdd1562cb1043582744b511f0e9730efdc489e891666068"},"schema_version":"1.0","source":{"id":"1512.08348","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.08348","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"arxiv_version","alias_value":"1512.08348v5","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.08348","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"pith_short_12","alias_value":"2J7LZXUKKZOD","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"2J7LZXUKKZOD74QH","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"2J7LZXUK","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:acf251973441858b811b4c925e29f491f726b9306f01a9b73f5c1df6f09f10c0","target":"graph","created_at":"2026-05-17T23:42:35Z","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":"The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman--Grassl correspondence, respectively.\n  The main new results are two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting rev","authors_text":"Alejandro Morales, Greta Panova, Igor Pak","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-28T08:53:22Z","title":"Hook formulas for skew shapes I. $q$-analogues and bijections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.08348","kind":"arxiv","version":5},"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":"sha256:4ec006a0e865679f8701eccf11234826facd2ed0e7cb40093009d3cf53efdcf9","target":"record","created_at":"2026-05-17T23:42:35Z","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":"d8115f4fa0956621bbf30c70a377c2136a83e502920531a00f9d9a9e73906978","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-28T08:53:22Z","title_canon_sha256":"1f646e6c9870b8224cdd1562cb1043582744b511f0e9730efdc489e891666068"},"schema_version":"1.0","source":{"id":"1512.08348","kind":"arxiv","version":5}},"canonical_sha256":"d27ebcde8a565c3ff207eba970c690fe8a58495688f355d82e5eb858604a7320","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d27ebcde8a565c3ff207eba970c690fe8a58495688f355d82e5eb858604a7320","first_computed_at":"2026-05-17T23:42:35.297659Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:35.297659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u08Ku8G6wAyG3OILYY1s22ZuNCv4o+wp8CqoXQYLWGDd6IKmHFX9EI80LY6h5d0KkJm9fXY3uwg3+ksWWZWVBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:35.298474Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.08348","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4ec006a0e865679f8701eccf11234826facd2ed0e7cb40093009d3cf53efdcf9","sha256:acf251973441858b811b4c925e29f491f726b9306f01a9b73f5c1df6f09f10c0"],"state_sha256":"df79cc8e8d0b0c87b11ffd2d23902fd240e9a3f28d186de5e61f92c442a4984f"}