{"title":"关于与分区曲柄相关的 1 的总数 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":"{\"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}","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
摘要
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。
On the total number of ones associated with cranks of partitions modulo 11
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 .
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