{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:C7DEYQSJQKK7HWWRYL4OZUBMWR","short_pith_number":"pith:C7DEYQSJ","schema_version":"1.0","canonical_sha256":"17c64c42498295f3dad1c2f8ecd02cb4794c09751a4cf5acb839a7ac42c96932","source":{"kind":"arxiv","id":"2110.03108","version":2},"attestation_state":"computed","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{"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2110.03108","created_at":"2026-05-25T02:00:59.746811+00:00"},{"alias_kind":"arxiv_version","alias_value":"2110.03108v2","created_at":"2026-05-25T02:00:59.746811+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.03108","created_at":"2026-05-25T02:00:59.746811+00:00"},{"alias_kind":"pith_short_12","alias_value":"C7DEYQSJQKK7","created_at":"2026-05-25T02:00:59.746811+00:00"},{"alias_kind":"pith_short_16","alias_value":"C7DEYQSJQKK7HWWR","created_at":"2026-05-25T02:00:59.746811+00:00"},{"alias_kind":"pith_short_8","alias_value":"C7DEYQSJ","created_at":"2026-05-25T02:00:59.746811+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR","json":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR.json","graph_json":"https://pith.science/api/pith-number/C7DEYQSJQKK7HWWRYL4OZUBMWR/graph.json","events_json":"https://pith.science/api/pith-number/C7DEYQSJQKK7HWWRYL4OZUBMWR/events.json","paper":"https://pith.science/paper/C7DEYQSJ"},"agent_actions":{"view_html":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR","download_json":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR.json","view_paper":"https://pith.science/paper/C7DEYQSJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2110.03108&json=true","fetch_graph":"https://pith.science/api/pith-number/C7DEYQSJQKK7HWWRYL4OZUBMWR/graph.json","fetch_events":"https://pith.science/api/pith-number/C7DEYQSJQKK7HWWRYL4OZUBMWR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/action/storage_attestation","attest_author":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/action/author_attestation","sign_citation":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/action/citation_signature","submit_replication":"https://pith.science/pith/C7DEYQSJQKK7HWWRYL4OZUBMWR/action/replication_record"}},"created_at":"2026-05-25T02:00:59.746811+00:00","updated_at":"2026-05-25T02:00:59.746811+00:00"}