Combinatorial interpretations of truncated series from the Jacobi triple product identity

IF 0.9 3区 数学 Q1 MATHEMATICS
Olivia X.M. Yao
{"title":"Combinatorial interpretations of truncated series from the Jacobi triple product identity","authors":"Olivia X.M. Yao","doi":"10.1016/j.ejc.2025.104176","DOIUrl":null,"url":null,"abstract":"<div><div>In their seminal work on truncated sums of theta series, Andrews and Merca, and Guo and Zeng independently posed a conjecture on the sign of the coefficients of truncated sums of Jacobi’s triple product identity. This conjecture was confirmed independently by Mao by utilizing analytic method and by Yee by using combinatorial method. In 2019, Wang and Yee reproved this conjecture by establishing an explicit series with nonnegative coefficients and proved a companion identity. In this paper, we present partition-theoretic interpretations of the truncated sums posed by Andrews and Merca and Guo and Zeng, and Wang and Yee. We determine what partitions of <span><math><mi>n</mi></math></span> are counted by the truncated sums based on the minimal excludant in congruence classes of partitions. As applications, we settle two open problems on partition-theoretic interpretations of series posed by Guo and Zeng, and Merca, respectively.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104176"},"PeriodicalIF":0.9000,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669825000617","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

In their seminal work on truncated sums of theta series, Andrews and Merca, and Guo and Zeng independently posed a conjecture on the sign of the coefficients of truncated sums of Jacobi’s triple product identity. This conjecture was confirmed independently by Mao by utilizing analytic method and by Yee by using combinatorial method. In 2019, Wang and Yee reproved this conjecture by establishing an explicit series with nonnegative coefficients and proved a companion identity. In this paper, we present partition-theoretic interpretations of the truncated sums posed by Andrews and Merca and Guo and Zeng, and Wang and Yee. We determine what partitions of n are counted by the truncated sums based on the minimal excludant in congruence classes of partitions. As applications, we settle two open problems on partition-theoretic interpretations of series posed by Guo and Zeng, and Merca, respectively.
Jacobi三重积恒等式中截短级数的组合解释
在关于级数截断和的开创性工作中,Andrews和Merca, Guo和Zeng分别对Jacobi的三重积单位的截断和的系数符号提出了一个猜想。这一猜想分别由Mao用解析法和Yee用组合法独立证实。2019年,Wang和Yee通过建立一个非负系数的显式级数,证明了一个伴恒等式,反驳了这一猜想。本文对Andrews和Merca、Guo和Zeng、Wang和Yee提出的截断和给出了分划理论解释。在划分的同余类中,我们根据最小不相容来确定n的哪些划分是由截断和计算的。作为应用,我们分别解决了Guo、Zeng和Merca提出的级数的划分理论解释的两个开放问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
2.10
自引率
10.00%
发文量
124
审稿时长
4-8 weeks
期刊介绍: The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信