{"paper":{"title":"Nearly Equal Distributions of the Rank and the Crank of Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Kathy Q. Ji, Wenston J.T. Zang, William Y.C. Chen","submitted_at":"2017-04-04T06:02:58Z","abstract_excerpt":"Let $N(\\leq m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(\\leq m,n)$ denote the number of partitions of $n$ with crank not greater than $m$. Bringmann and Mahlburg observed that $N(\\leq m,n)\\leq M(\\leq m,n)\\leq N(\\leq m+1,n)$ for $m<0$ and $1\\leq n\\leq 100$. They also pointed out that these inequalities can be restated as the existence of a re-ordering $\\tau_n$ on the set of partitions of $n$ such that $|\\text{crank}(\\lambda)|-|\\text{rank}(\\tau_n(\\lambda))|=0$ or $1$ for all partitions $\\lambda$ of $n$, that is, the rank and the crank are nearly equal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.00882","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":""},"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"}