{"title":"Overpartitions in terms of 2-adic valuation","authors":"Mircea Merca","doi":"10.1007/s00010-024-01117-6","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we consider the 2-adic valuation of integers and provide an alternative representation for the generating function of the number of overpartitions of an integer. As a consequence of this result, we obtain a new formula and a new combinatorial interpretation for the number of overpartitions of an integer. This formula implies a certain type of partitions with restrictions for which we provide two Ramanujan-type congruences and present as open problems two infinite families of linear inequalities. Connections between overpartitions and the game of <i>m</i>-Modular Nim with two heaps are presented in this context.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Aequationes Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00010-024-01117-6","RegionNum":3,"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 this paper, we consider the 2-adic valuation of integers and provide an alternative representation for the generating function of the number of overpartitions of an integer. As a consequence of this result, we obtain a new formula and a new combinatorial interpretation for the number of overpartitions of an integer. This formula implies a certain type of partitions with restrictions for which we provide two Ramanujan-type congruences and present as open problems two infinite families of linear inequalities. Connections between overpartitions and the game of m-Modular Nim with two heaps are presented in this context.
期刊介绍：
aequationes mathematicae is an international journal of pure and applied mathematics, which emphasizes functional equations, dynamical systems, iteration theory, combinatorics, and geometry. The journal publishes research papers, reports of meetings, and bibliographies. High quality survey articles are an especially welcome feature. In addition, summaries of recent developments and research in the field are published rapidly.
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