{"paper":{"title":"Chain Decompositions of $q,t$-Catalan Numbers via Local Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kyungyong Lee, Li Li, Nicholas A. Loehr, Seongjune Han","submitted_at":"2020-03-09T02:43:19Z","abstract_excerpt":"The $q,t$-Catalan number $\\mathrm{Cat}_n(q,t)$ enumerates integer partitions contained in an $n\\times n$ triangle by their dinv and external area statistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))] proposed a new approach to understanding the symmetry property $\\mathrm{Cat}_n(q,t)=\\mathrm{Cat}_n(t,q)$ based on decomposing the set of all integer partitions into infinite chains. Each such global chain $\\mathcal{C}_{\\mu}$ has an opposite chain $\\mathcal{C}_{\\mu^*}$; these combine to give a new small slice of $\\mathrm{Cat}_n(q,t)$ that is symmetric in $q$ and $t$. Here "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.03896","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/2003.03896/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"}