{"paper":{"title":"Schubert puzzles and integrability III: separated descents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Allen Knutson, Paul Zinn-Justin","submitted_at":"2023-06-24T04:06:55Z","abstract_excerpt":"In paper I of this series we gave positive formulae for expanding the product $\\mathfrak S^\\pi \\mathfrak S^\\rho$ of two Schubert polynomials, in the case that both $\\pi,\\rho$ had shared descent set of size $\\leq 3$. Here we introduce and give positive formulae for two new classes of Schubert product problems: separated descent in which $\\pi$'s last descent occurs at (or before) $\\rho$'s first, and almost separated descent in which $\\pi$'s last two descents occur at (or before) $\\rho$'s first two respectively. In both cases our puzzle formulae extend to $K$-theory (multiplying Grothendieck poly"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.13855","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2306.13855/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"}