{"paper":{"title":"Trivial Meet and Join within the Lattice of Monotone Triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Hammett, John Engbers","submitted_at":"2014-01-21T18:17:43Z","abstract_excerpt":"The lattice of monotone triangles $(\\mathfrak{M}_n,\\le)$ ordered by entry-wise comparisons is studied. Let $\\tau_{\\min}$ denote the unique minimal element in this lattice, and $\\tau_{\\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(\\tau_1,\\ldots,\\tau_r)$ with minimal infimum $\\tau_{\\min}$ (maximal supremum $\\tau_{\\max}$, resp.) is shown to asymptotically approach $r|\\mathfrak{M}_n|^{r-1}$ as $n \\to \\infty$. Thus, with high probability this event implies that one of the $\\tau_i$ is $\\tau_{\\min}$ ($\\tau_{\\max}$, resp.). Higher-order error terms are also discussed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5409","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":""},"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"}