{"paper":{"title":"Biases among Congruence Classes for Parts in k-regular Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Faye Jackson, Misheel Otgonbayar","submitted_at":"2022-07-10T00:38:37Z","abstract_excerpt":"For integers $k,t \\geq 2$ and $1\\leq r \\leq t$ let $D_k(r,t;n)$ be the number of parts among all $k$-regular partitions (i.e., partitions of $n$ where all parts have multiplicity less than $k$) of $n$ that are congruent to $r$ modulo $t$. Using the circle method, we obtain the asymptotic\n  \\[\n  D_{k}(r,t;n) = \\frac{3^{\\frac{1}{4}}e^{\\pi\\sqrt{\\frac{2Kn}{3}}}}{\\pi t 2^{\\frac{3}{4}}K^{\\frac{1}{4}}n^{\\frac{1}{4}}\\sqrt{k}}\\left(\\log k + \\left(\\frac{3\\sqrt{K}\\log k}{8\\sqrt{6}\\pi} - \\frac{t\\pi(k-1)K^{\\frac{1}{2}}}{2\\sqrt{6}}\\left(\\frac{r}{t}- \\frac{1}{2}\\right)\\right)n^{-\\frac{1}{2}} + O(n^{-1})\\righ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.04352","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/2207.04352/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"}