{"paper":{"title":"Counting partitions inside a rectangle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.PR"],"primary_cat":"math.CO","authors_text":"Greta Panova, Robin Pemantle, Stephen Melczer","submitted_at":"2018-05-22T03:35:23Z","abstract_excerpt":"We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \\times \\ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\\binom{m+\\ell}{m}_q$. We obtain sharp asymptotics throughout the regime $\\ell = \\Theta (m)$ and $n = \\Theta (m^2)$. Previously, sharp asymptotics were derived by Tak\\'acs only in the regime where $|n - \\ell m /2| = O(\\sqrt{\\ell m (\\ell + m)})$ using a local central limit theorem. Our approach is to solve a related large deviation problem: we describe the tilted measure that produces configurations whose bounding recta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.08375","kind":"arxiv","version":3},"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"}