{"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"}