{"title":"Symmetric polynomials associated with numerical semigroups","authors":"L. Fel","doi":"10.47443/dml.2020.0066","DOIUrl":null,"url":null,"abstract":"We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums E_k=\\sum_{j=1}^m x_j^k. We observe a visual similarity between normalized polynomials P_n(x_1,...,x_m)/\\chi_m, where \\chi_m=\\prod_{j=1}^m x_j, and a polynomial part of a partition function W(s,{d_1,...,d_m}), which gives a number of partitions of s\\ge 0 into m positive integers d_j, and put forward a conjecture about their relationship.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"136 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.47443/dml.2020.0066","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums E_k=\sum_{j=1}^m x_j^k. We observe a visual similarity between normalized polynomials P_n(x_1,...,x_m)/\chi_m, where \chi_m=\prod_{j=1}^m x_j, and a polynomial part of a partition function W(s,{d_1,...,d_m}), which gives a number of partitions of s\ge 0 into m positive integers d_j, and put forward a conjecture about their relationship.