{"title":"Two conjectures of Andrews, Merca and Yee on truncated theta series","authors":"Shane Chern , Ernest X.W. Xia","doi":"10.1016/j.jcta.2024.105874","DOIUrl":null,"url":null,"abstract":"<div><p>In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> to count the number of partitions of <em>n</em> in which <em>k</em> is the least integer that is not a part and there are more parts exceeding <em>k</em> than there are below <em>k</em>. In recent years, two conjectures concerning <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large <em>n</em> whenever <em>k</em> is fixed.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S009731652400013X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function to count the number of partitions of n in which k is the least integer that is not a part and there are more parts exceeding k than there are below k. In recent years, two conjectures concerning on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large n whenever k is fixed.
安德鲁斯和梅尔卡在研究欧拉五边形数截断定理时，引入了一个分区函数 Mk(n)，用来计算 n 的分区数，其中 k 是不属于分区的最小整数，且超过 k 的分区数多于低于 k 的分区数。近年来，安德鲁斯、梅尔卡和易提出了关于截断θ数列 Mk(n) 的两个猜想。在本文中，我们证明了只要 k 固定不变，对于足够大的 n，这两个猜想都是真的。
期刊介绍：
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.
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