{"title":"On the total number of ones associated with cranks of partitions modulo 11","authors":"Dandan Chen , Rong Chen , Siyu Yin","doi":"10.1016/j.jmaa.2024.128954","DOIUrl":null,"url":null,"abstract":"<div><div>In 2021, Andrews mentioned that George Beck introduced partition statistics <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, which denote the total number of ones in the partition of <em>n</em> with crank congruent to <em>r</em> modulo <em>m</em>. Recently, a number of congruences and identities involving <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> for some small <em>m</em> have been developed. We establish the 11-dissection of the generating functions for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>,</mo><mn>11</mn><mo>,</mo><mi>n</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>(</mo><mn>11</mn><mo>−</mo><mi>r</mi><mo>,</mo><mn>11</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, where <span><math><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></math></span>. In particular, we discover a beautiful identity involving <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>,</mo><mn>11</mn><mo>,</mo><mn>11</mn><mi>n</mi><mo>+</mo><mn>6</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 1","pages":"Article 128954"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X2400876X","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 2021, Andrews mentioned that George Beck introduced partition statistics , which denote the total number of ones in the partition of n with crank congruent to r modulo m. Recently, a number of congruences and identities involving for some small m have been developed. We establish the 11-dissection of the generating functions for , where . In particular, we discover a beautiful identity involving .
2021 年,安德鲁斯提到乔治-贝克引入了分部统计量 Mw(r,m,n),表示 n 的分部中曲柄与 r 同调的 m 的总数。我们建立了 Mω(r,11,n)-Mω(11-r,11,n)(其中 r=1,2,3,4,5)生成函数的 11 分段。我们特别发现了一个涉及 Mω(r,11,11n+6)的美丽特性。
期刊介绍:
The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.
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