{"title":"Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions.","authors":"Michael J Schlosser, Nian Hong Zhou","doi":"10.1007/s40687-025-00502-0","DOIUrl":null,"url":null,"abstract":"<p><p>We prove specific biases in the number of occurrences of parts belonging to two different residue classes <i>a</i> and <i>b</i>, modulo a fixed nonnegative integer <i>m</i>, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size <i>n</i> that belong to these sets of partitions and have a symmetric residue class bias (i.e., for <math><mrow><mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo><</mo> <mi>m</mi> <mo>/</mo> <mn>2</mn></mrow> </math> and <math><mrow><mi>b</mi> <mo>=</mo> <mi>m</mi> <mo>-</mo> <mi>a</mi></mrow> </math> ), as <i>n</i> tends to infinity.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 1","pages":"17"},"PeriodicalIF":1.2000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11845404/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Research in the Mathematical Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40687-025-00502-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/2/21 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove specific biases in the number of occurrences of parts belonging to two different residue classes a and b, modulo a fixed nonnegative integer m, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size n that belong to these sets of partitions and have a symmetric residue class bias (i.e., for and ), as n tends to infinity.
我们证明了对于无限制划分集、划分成不同部分集和过划分集，模于固定非负整数m，属于两个不同剩余类a和b的部分出现次数的特定偏差。这些偏差来自于残差加权配分函数的不平等。当n趋于无穷时，我们还建立了大小为n的分区数目的渐近公式，这些分区属于这些分区集，并且具有对称剩余类偏差（即，对于1≤a m / 2和b = m - a）。
期刊介绍：
Research in the Mathematical Sciences is an international, peer-reviewed hybrid journal covering the full scope of Theoretical Mathematics, Applied Mathematics, and Theoretical Computer Science. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to the research areas of both theoretical and applied mathematics and theoretical computer science.
This journal is an efficient enterprise where the editors play a central role in soliciting the best research papers, and where editorial decisions are reached in a timely fashion. Research in the Mathematical Sciences does not have a length restriction and encourages the submission of longer articles in which more complex and detailed analysis and proofing of theorems is required. It also publishes shorter research communications (Letters) covering nascent research in some of the hottest areas of mathematical research. This journal will publish the highest quality papers in all of the traditional areas of applied and theoretical areas of mathematics and computer science, and it will actively seek to publish seminal papers in the most emerging and interdisciplinary areas in all of the mathematical sciences. Research in the Mathematical Sciences wishes to lead the way by promoting the highest quality research of this type.
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