{"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/parts-in-k-indivisible-partitions-always","title":"Parts in $k$-indivisible Partitions Always Display Biases between Residue Classes","arxiv_id":"2305.05753","date":"2023-05-09","proceeding":null,"authors":["Faye Jackson","Misheel Otgonbayar"],"abstract":"Let $k, t$ be coprime integers, and let $1 \\leq r \\leq t$. We let $D_k^\\times(r,t;n)$ denote the total number of parts among all $k$-indivisible partitions (i.e., those partitions where no part is divisible by $k$) of $n$ which are congruent to $r$ modulo $t$. In previous work of the authors, an asymptotic estimate for $D_k^\\times(r,t;n)$ was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture that there are no \"ties\" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of $L$-functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing to conclude that there is always a bias towards one congruence class or another modulo $t$ among all parts in $k$-indivisible partitions of $n$ as $n$ becomes large.","url_abs":"https://arxiv.org/abs/2305.05753v1","url_pdf":"https://arxiv.org/pdf/2305.05753v1.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":"parts-in-k-indivisible-partitions-always","repo_url":"https://github.com/FayeAlephNil/KIndivisibleBiases","is_official":1,"mentioned_in_paper":0,"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}