{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/biases-among-congruence-classes-for-parts-in","title":"Biases among Congruence Classes for Parts in k-regular Partitions","arxiv_id":"2207.04352","date":"2022-07-10","proceeding":null,"authors":["Faye Jackson","Misheel Otgonbayar"],"abstract":"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 \\[ 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})\\right), \\] where $K = 1 - \\frac{1}{k}$. The main term of this asymptotic does not depend on $r$, and so if $P_k(n)$ is the total number of parts among all $k$-regular partitions of $n$, we have that $\\frac{D_k(r,t;n)}{P_k(n)} \\to \\frac{1}{t}$ as $n \\to \\infty$. Thus, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards the lower congruence classes; that is, for $1\\leq r < s \\leq t$ we have $D_k(r,t;n) \\geq D_k(s,t;n)$ for sufficiently large $n$. We make this inequality explicit, showing that for $3 \\leq k \\leq 10$ and $2 \\leq t \\leq 10$ the inequality $D_k(r,t;n) \\geq D_k(s,t;n)$ holds for all $n \\geq 1$ and the strict inequality $D_k(r,t;n) > D_k(s,t;n)$ holds for all $n \\geq 17$.","url_abs":"https://arxiv.org/abs/2207.04352v1","url_pdf":"https://arxiv.org/pdf/2207.04352v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"biases-among-congruence-classes-for-parts-in","repo_url":"https://github.com/fayealephnil/kregularbiases","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}