{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:FDZABLGFBAPFXUGP5DA4LB37VZ","short_pith_number":"pith:FDZABLGF","canonical_record":{"source":{"id":"1203.4729","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-21T13:30:08Z","cross_cats_sorted":[],"title_canon_sha256":"81c0bbe694a216ad0fc87be2cba63862b7ad105fed8980e6b1f825c3cf5b186f","abstract_canon_sha256":"c014495ab1b7f39ecd8fca202aacd07629618030b63c6d00f8e4c5a8e75c9610"},"schema_version":"1.0"},"canonical_sha256":"28f200acc5081e5bd0cfe8c1c5877fae4efe6f4b5ae31ab9876829deeaa407b3","source":{"kind":"arxiv","id":"1203.4729","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.4729","created_at":"2026-05-18T03:59:37Z"},{"alias_kind":"arxiv_version","alias_value":"1203.4729v1","created_at":"2026-05-18T03:59:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.4729","created_at":"2026-05-18T03:59:37Z"},{"alias_kind":"pith_short_12","alias_value":"FDZABLGFBAPF","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FDZABLGFBAPFXUGP","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FDZABLGF","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:FDZABLGFBAPFXUGP5DA4LB37VZ","target":"record","payload":{"canonical_record":{"source":{"id":"1203.4729","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-21T13:30:08Z","cross_cats_sorted":[],"title_canon_sha256":"81c0bbe694a216ad0fc87be2cba63862b7ad105fed8980e6b1f825c3cf5b186f","abstract_canon_sha256":"c014495ab1b7f39ecd8fca202aacd07629618030b63c6d00f8e4c5a8e75c9610"},"schema_version":"1.0"},"canonical_sha256":"28f200acc5081e5bd0cfe8c1c5877fae4efe6f4b5ae31ab9876829deeaa407b3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:37.347592Z","signature_b64":"rtlWYMu3m75qfjFSygY6zLE+MxaheQIdGO9u6w1L+0Orju7KO440mtmXjRDk/OtH0OfAKdj0ux2rQsGHI4WmBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28f200acc5081e5bd0cfe8c1c5877fae4efe6f4b5ae31ab9876829deeaa407b3","last_reissued_at":"2026-05-18T03:59:37.346871Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:37.346871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.4729","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:59:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EmPSdcTMeXaYPP4aWtXPpRZuTCm391aWSYg4ISGZ3cpzl43+aU8RvXcBRnBS1yzRcs7um6RL8O9eaGT1k/lDCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T08:45:51.911093Z"},"content_sha256":"8268557d17d409c9216a5f6e23ae5a7f2f21534038d570d4bb0d566a90f20eed","schema_version":"1.0","event_id":"sha256:8268557d17d409c9216a5f6e23ae5a7f2f21534038d570d4bb0d566a90f20eed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:FDZABLGFBAPFXUGP5DA4LB37VZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Raising operators and the Littlewood-Richardson polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Fun","submitted_at":"2012-03-21T13:30:08Z","abstract_excerpt":"We use Young's raising operators to derive a Pieri rule for the ring generated by the indeterminates $h_{r,s}$ given in Macdonald's 9th Variation of the Schur functions. Under an appropriate specialisation of $h_{r,s}$, we derive the Pieri rule for the ring $\\La(a)$ of double symmetric functions, which has a basis consisting of the double Schur functions. Together with a suitable interpretation of the Jacobi--Trudi identity, our Pieri rule allows us to obtain a new proof of a rule to calculate the Littlewood--Richardson polynomials, which gives a multiplication rule for the double Schur functi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.4729","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:59:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lvoBT56x1iIiJXcT+DZElEfCCrt1ChLsfTdc4NFegK3LspS7n3hiKJhrRD/UNQVBvYASI0Sqm723NnOFperHCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T08:45:51.911399Z"},"content_sha256":"36de268f95fbe338f737d59f270ba312d51e190230e2f0bfda78c503425c994b","schema_version":"1.0","event_id":"sha256:36de268f95fbe338f737d59f270ba312d51e190230e2f0bfda78c503425c994b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FDZABLGFBAPFXUGP5DA4LB37VZ/bundle.json","state_url":"https://pith.science/pith/FDZABLGFBAPFXUGP5DA4LB37VZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FDZABLGFBAPFXUGP5DA4LB37VZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-22T08:45:51Z","links":{"resolver":"https://pith.science/pith/FDZABLGFBAPFXUGP5DA4LB37VZ","bundle":"https://pith.science/pith/FDZABLGFBAPFXUGP5DA4LB37VZ/bundle.json","state":"https://pith.science/pith/FDZABLGFBAPFXUGP5DA4LB37VZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FDZABLGFBAPFXUGP5DA4LB37VZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FDZABLGFBAPFXUGP5DA4LB37VZ","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":"c014495ab1b7f39ecd8fca202aacd07629618030b63c6d00f8e4c5a8e75c9610","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-21T13:30:08Z","title_canon_sha256":"81c0bbe694a216ad0fc87be2cba63862b7ad105fed8980e6b1f825c3cf5b186f"},"schema_version":"1.0","source":{"id":"1203.4729","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.4729","created_at":"2026-05-18T03:59:37Z"},{"alias_kind":"arxiv_version","alias_value":"1203.4729v1","created_at":"2026-05-18T03:59:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.4729","created_at":"2026-05-18T03:59:37Z"},{"alias_kind":"pith_short_12","alias_value":"FDZABLGFBAPF","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FDZABLGFBAPFXUGP","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FDZABLGF","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:36de268f95fbe338f737d59f270ba312d51e190230e2f0bfda78c503425c994b","target":"graph","created_at":"2026-05-18T03:59:37Z","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 use Young's raising operators to derive a Pieri rule for the ring generated by the indeterminates $h_{r,s}$ given in Macdonald's 9th Variation of the Schur functions. Under an appropriate specialisation of $h_{r,s}$, we derive the Pieri rule for the ring $\\La(a)$ of double symmetric functions, which has a basis consisting of the double Schur functions. Together with a suitable interpretation of the Jacobi--Trudi identity, our Pieri rule allows us to obtain a new proof of a rule to calculate the Littlewood--Richardson polynomials, which gives a multiplication rule for the double Schur functi","authors_text":"Alex Fun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-21T13:30:08Z","title":"Raising operators and the Littlewood-Richardson polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.4729","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":"sha256:8268557d17d409c9216a5f6e23ae5a7f2f21534038d570d4bb0d566a90f20eed","target":"record","created_at":"2026-05-18T03:59:37Z","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":"c014495ab1b7f39ecd8fca202aacd07629618030b63c6d00f8e4c5a8e75c9610","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-21T13:30:08Z","title_canon_sha256":"81c0bbe694a216ad0fc87be2cba63862b7ad105fed8980e6b1f825c3cf5b186f"},"schema_version":"1.0","source":{"id":"1203.4729","kind":"arxiv","version":1}},"canonical_sha256":"28f200acc5081e5bd0cfe8c1c5877fae4efe6f4b5ae31ab9876829deeaa407b3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"28f200acc5081e5bd0cfe8c1c5877fae4efe6f4b5ae31ab9876829deeaa407b3","first_computed_at":"2026-05-18T03:59:37.346871Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:37.346871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rtlWYMu3m75qfjFSygY6zLE+MxaheQIdGO9u6w1L+0Orju7KO440mtmXjRDk/OtH0OfAKdj0ux2rQsGHI4WmBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:37.347592Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.4729","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8268557d17d409c9216a5f6e23ae5a7f2f21534038d570d4bb0d566a90f20eed","sha256:36de268f95fbe338f737d59f270ba312d51e190230e2f0bfda78c503425c994b"],"state_sha256":"077e95010fe5a8103d02fab5cd468889dc383a5064eb48199e543f92e3e92704"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RP1yb9EJCMh/OZ7iLyeZDIXHJzHYuO1T7WK6dczONV6/fEPL/MURmcWNaPCERIwTdXYLS8fIqo5zXVKfBwTyDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T08:45:51.914391Z","bundle_sha256":"90b2aba008ef10fdc5387ee3c9db3423e8701a7314ae0d1c7ce06440afb77b44"}}