{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:C7DEYQSJQKK7HWWRYL4OZUBMWR","short_pith_number":"pith:C7DEYQSJ","canonical_record":{"source":{"id":"2110.03108","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2021-10-06T23:48:04Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"098583fbee618ea79dca6e13ac4f915aa9ebdb9c2fef9026223ecd3fdee0c972","abstract_canon_sha256":"a39b72ad09daac9bfcc0523a88e36c18d25e3d9c944359fda5a728da30832597"},"schema_version":"1.0"},"canonical_sha256":"17c64c42498295f3dad1c2f8ecd02cb4794c09751a4cf5acb839a7ac42c96932","source":{"kind":"arxiv","id":"2110.03108","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.03108","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"arxiv_version","alias_value":"2110.03108v2","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.03108","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"pith_short_12","alias_value":"C7DEYQSJQKK7","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"pith_short_16","alias_value":"C7DEYQSJQKK7HWWR","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"pith_short_8","alias_value":"C7DEYQSJ","created_at":"2026-05-25T02:00:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:C7DEYQSJQKK7HWWRYL4OZUBMWR","target":"record","payload":{"canonical_record":{"source":{"id":"2110.03108","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2021-10-06T23:48:04Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"098583fbee618ea79dca6e13ac4f915aa9ebdb9c2fef9026223ecd3fdee0c972","abstract_canon_sha256":"a39b72ad09daac9bfcc0523a88e36c18d25e3d9c944359fda5a728da30832597"},"schema_version":"1.0"},"canonical_sha256":"17c64c42498295f3dad1c2f8ecd02cb4794c09751a4cf5acb839a7ac42c96932","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:00:59.747464Z","signature_b64":"fGQmn1BtKPPvD0N+WYntWvCTNJfXuuAM1ab19CWnjzZ6b8mrfHmKk31G4eV9arWZqHwMdYR6QVeIm57ZQkVCBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17c64c42498295f3dad1c2f8ecd02cb4794c09751a4cf5acb839a7ac42c96932","last_reissued_at":"2026-05-25T02:00:59.746699Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:00:59.746699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2110.03108","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-25T02:00:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CnvZJHlluxwSP/5ub1jIm1RTP0B+aw9yxdyaB17kNRkgFPwwVeTy3jBnmoAUjLNOee9oG1flnF3fHReMa9RVAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:55:03.778094Z"},"content_sha256":"71ee8edf93c7c9392a17797dae500067c7869d9d8d77b00bc3390a9c129b58ce","schema_version":"1.0","event_id":"sha256:71ee8edf93c7c9392a17797dae500067c7869d9d8d77b00bc3390a9c129b58ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:C7DEYQSJQKK7HWWRYL4OZUBMWR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The pre-Pieri rules","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CO","authors_text":"Darij Grinberg","submitted_at":"2021-10-06T23:48:04Z","abstract_excerpt":"Let $R$ be a commutative ring and $n\\geq1$ and $p\\geq0$ two integers. Let $h_{k,\\ i}$ be an element of $R$ for all $k\\in\\mathbb Z$ and $i\\in [n]$. For any $\\alpha\\in\\mathbb Z^n$, we define \\[ t_{\\alpha}:=\\det\\begin{pmatrix} h_{\\alpha_1+1,\\ 1} & h_{\\alpha_1+2,\\ 1} & \\cdots & h_{\\alpha_1+n,\\ 1}\\\\ h_{\\alpha_2+1,\\ 2} & h_{\\alpha_2+2,\\ 2} & \\cdots & h_{\\alpha_2+n,\\ 2}\\\\ \\vdots & \\vdots & \\ddots & \\vdots\\\\ h_{\\alpha_n+1,\\ n} & h_{\\alpha_n+2,\\ n} & \\cdots & h_{\\alpha_n+n,\\ n} \\end{pmatrix} \\in R \\] (where $\\alpha_i$ denotes the $i$-th entry of $\\alpha$). Then, we have the identity \\[ \\sum_{\\substack{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.03108","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2110.03108/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-25T02:00:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V9BpqEV09k9lrnLzzehA3VJbm790YqNyAgZLXI5X4NpU+CnGO/rmvVyc1IhGVTIf3Ul5IDantN+DNcrn4q9fAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:55:03.778507Z"},"content_sha256":"c0510412242e3b242080c86f23aba70269a1371689387faf5c0b0d03739bbcac","schema_version":"1.0","event_id":"sha256:c0510412242e3b242080c86f23aba70269a1371689387faf5c0b0d03739bbcac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/bundle.json","state_url":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/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":"2026-05-28T05:55:03Z","links":{"resolver":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR","bundle":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/bundle.json","state":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:C7DEYQSJQKK7HWWRYL4OZUBMWR","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":"a39b72ad09daac9bfcc0523a88e36c18d25e3d9c944359fda5a728da30832597","cross_cats_sorted":["math.RA"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2021-10-06T23:48:04Z","title_canon_sha256":"098583fbee618ea79dca6e13ac4f915aa9ebdb9c2fef9026223ecd3fdee0c972"},"schema_version":"1.0","source":{"id":"2110.03108","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.03108","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"arxiv_version","alias_value":"2110.03108v2","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.03108","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"pith_short_12","alias_value":"C7DEYQSJQKK7","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"pith_short_16","alias_value":"C7DEYQSJQKK7HWWR","created_at":"2026-05-25T02:00:59Z"},{"alias_kind":"pith_short_8","alias_value":"C7DEYQSJ","created_at":"2026-05-25T02:00:59Z"}],"graph_snapshots":[{"event_id":"sha256:c0510412242e3b242080c86f23aba70269a1371689387faf5c0b0d03739bbcac","target":"graph","created_at":"2026-05-25T02:00:59Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2110.03108/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $R$ be a commutative ring and $n\\geq1$ and $p\\geq0$ two integers. Let $h_{k,\\ i}$ be an element of $R$ for all $k\\in\\mathbb Z$ and $i\\in [n]$. For any $\\alpha\\in\\mathbb Z^n$, we define \\[ t_{\\alpha}:=\\det\\begin{pmatrix} h_{\\alpha_1+1,\\ 1} & h_{\\alpha_1+2,\\ 1} & \\cdots & h_{\\alpha_1+n,\\ 1}\\\\ h_{\\alpha_2+1,\\ 2} & h_{\\alpha_2+2,\\ 2} & \\cdots & h_{\\alpha_2+n,\\ 2}\\\\ \\vdots & \\vdots & \\ddots & \\vdots\\\\ h_{\\alpha_n+1,\\ n} & h_{\\alpha_n+2,\\ n} & \\cdots & h_{\\alpha_n+n,\\ n} \\end{pmatrix} \\in R \\] (where $\\alpha_i$ denotes the $i$-th entry of $\\alpha$). Then, we have the identity \\[ \\sum_{\\substack{","authors_text":"Darij Grinberg","cross_cats":["math.RA"],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2021-10-06T23:48:04Z","title":"The pre-Pieri rules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.03108","kind":"arxiv","version":2},"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":"sha256:71ee8edf93c7c9392a17797dae500067c7869d9d8d77b00bc3390a9c129b58ce","target":"record","created_at":"2026-05-25T02:00:59Z","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":"a39b72ad09daac9bfcc0523a88e36c18d25e3d9c944359fda5a728da30832597","cross_cats_sorted":["math.RA"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2021-10-06T23:48:04Z","title_canon_sha256":"098583fbee618ea79dca6e13ac4f915aa9ebdb9c2fef9026223ecd3fdee0c972"},"schema_version":"1.0","source":{"id":"2110.03108","kind":"arxiv","version":2}},"canonical_sha256":"17c64c42498295f3dad1c2f8ecd02cb4794c09751a4cf5acb839a7ac42c96932","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"17c64c42498295f3dad1c2f8ecd02cb4794c09751a4cf5acb839a7ac42c96932","first_computed_at":"2026-05-25T02:00:59.746699Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:00:59.746699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fGQmn1BtKPPvD0N+WYntWvCTNJfXuuAM1ab19CWnjzZ6b8mrfHmKk31G4eV9arWZqHwMdYR6QVeIm57ZQkVCBg==","signature_status":"signed_v1","signed_at":"2026-05-25T02:00:59.747464Z","signed_message":"canonical_sha256_bytes"},"source_id":"2110.03108","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71ee8edf93c7c9392a17797dae500067c7869d9d8d77b00bc3390a9c129b58ce","sha256:c0510412242e3b242080c86f23aba70269a1371689387faf5c0b0d03739bbcac"],"state_sha256":"f0260378296c27602b91cddebeb3227c04e64e3a75fba89fc2ec5d3c5b6061f4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6tssDCxLU/Ov4U27Hg9CKe8ILNooeULF8erbTIOAJ+wkr26arSl1Wo3LHTbvGF9j7BC3fiX54/hv/tKSWN5iDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T05:55:03.780796Z","bundle_sha256":"41e9f8056312fe0069258d60d7207b1551e080327242b62ff43b41655ad11cca"}}