{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:23HDCAE5CH2E4TXMRYIPBEPTMV","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":"b18d14033fbf39a5c67891dd689c1d163f12708fe4d64f4f833312abba92ce60","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-19T20:42:53Z","title_canon_sha256":"65f6c1eaddacb58ed4691e6aaab51aa4fc5d2635d2a5a214414ca864d42a6301"},"schema_version":"1.0","source":{"id":"1112.4460","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.4460","created_at":"2026-05-18T04:02:03Z"},{"alias_kind":"arxiv_version","alias_value":"1112.4460v3","created_at":"2026-05-18T04:02:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4460","created_at":"2026-05-18T04:02:03Z"},{"alias_kind":"pith_short_12","alias_value":"23HDCAE5CH2E","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"23HDCAE5CH2E4TXM","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"23HDCAE5","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:8d08088e17dd8db1ade23bf557a1ee9942a3fe04c8af8dc3b28a2d5cbf611a1a","target":"graph","created_at":"2026-05-18T04:02:03Z","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 prove that the Lam-Shimozono \"down operator\" on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of \"near rectangles\" in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood--Richardson coefficients.","authors_text":"Chris Berg, Franco Saliola, Luis Serrano","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-19T20:42:53Z","title":"The down operator and expansions of near rectangular k-Schur functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4460","kind":"arxiv","version":3},"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":"sha256:8c8f52de4f0627f1ec498f582b1a34802c5dbc09e15ce7b0156ed0361ecd240f","target":"record","created_at":"2026-05-18T04:02:03Z","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":"b18d14033fbf39a5c67891dd689c1d163f12708fe4d64f4f833312abba92ce60","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-19T20:42:53Z","title_canon_sha256":"65f6c1eaddacb58ed4691e6aaab51aa4fc5d2635d2a5a214414ca864d42a6301"},"schema_version":"1.0","source":{"id":"1112.4460","kind":"arxiv","version":3}},"canonical_sha256":"d6ce31009d11f44e4eec8e10f091f3657c5885757a75daf91554bdc7e3b1af0b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6ce31009d11f44e4eec8e10f091f3657c5885757a75daf91554bdc7e3b1af0b","first_computed_at":"2026-05-18T04:02:03.665657Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:02:03.665657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q2SN9fLZzMCbSiszWECjjwqy7ve/v2voZRwgHHXTorW1WsDaCEMTnM0m9z9/MTsp/WPgzC+5ySVrRIpcQAhNCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:02:03.666434Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.4460","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c8f52de4f0627f1ec498f582b1a34802c5dbc09e15ce7b0156ed0361ecd240f","sha256:8d08088e17dd8db1ade23bf557a1ee9942a3fe04c8af8dc3b28a2d5cbf611a1a"],"state_sha256":"bc2df93a722750d84a4343048d8bc7a1147d070971cd5a3fcad4d32b42fafc95"}