{"paper":{"title":"Exact thresholds for Schur positivity of the lattices $\\mathbf m\\times\\mathbf 2$ and $\\mathbf m\\times\\mathbf 3$","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David G.L. Wang, K. Zhang","submitted_at":"2025-10-03T15:43:22Z","abstract_excerpt":"We determine the exact thresholds for Schur positivity in the two families of chain products $\\mathbf m\\times\\mathbf2$ and $\\mathbf m\\times\\mathbf3$: the former is Schur positive exactly for $m\\le7$, and the latter exactly for $m\\le6$. For $m\\ge8$, we prove non-Schur-positivity in both families by exhibiting explicit negative Schur coefficients obtained from Pieri's rules and stable-composition counts. The remaining finite cases are settled by exact SageMath computations; in particular, $\\mathbf7\\times\\mathbf3$ has a negative Schur coefficient. These results settle the $n=2$ and $n=3$ cases in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.03116","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/2510.03116/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"}