安德鲁斯、梅尔卡和易关于截断θ级数的两个猜想

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2024-02-22 DOI:10.1016/j.jcta.2024.105874

Shane Chern , Ernest X.W. Xia

{"title":"安德鲁斯、梅尔卡和易关于截断θ级数的两个猜想","authors":"Shane Chern , Ernest X.W. Xia","doi":"10.1016/j.jcta.2024.105874","DOIUrl":null,"url":null,"abstract":"<div>In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function <math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> 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 <math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> 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.</div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105874"},"PeriodicalIF":0.9000,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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>In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function <math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> 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 <math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> 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.</div>\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":\"206 \",\"pages\":\"Article 105874\"},\"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}","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

摘要

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Two conjectures of Andrews, Merca and Yee on truncated theta series

In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function $M_{k} (n)$ 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 $M_{k} (n)$ 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.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of Combinatorial Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

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.