{"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 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.
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.