{"paper":{"title":"Weak order: Alternating sign matrices, monotone triangles, and bumpless pipe dreams","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anna Weigandt, Laura Escobar, Patricia Klein","submitted_at":"2026-06-11T16:09:05Z","abstract_excerpt":"In 2018, Hamaker and Reiner introduced weak order for monotone triangles, which extended the usual notion of weak order on the symmetric group. Monotone triangles on $\\{1, \\ldots, n\\}$ are well-known to be in bijection with the set ASM$(n)$ of $n \\times n$ alternating sign matrices. Hamaker and Reiner defined weak order on ASM$(n)$ to be induced from weak order on monotone triangles via the standard bijection. Recently, the present authors used an a priori different definition of weak order on ASM$(n)$ to give a combinatorial characterization of the codimension of ASM varieties and to show tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13518","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/2606.13518/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"}