{"title":"d 折分钻石","authors":"","doi":"10.1016/j.disc.2024.114163","DOIUrl":null,"url":null,"abstract":"<div><p>In this work we introduce new combinatorial objects called <em>d</em>–fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> to be their counting function. We also consider the Schmidt type <em>d</em>–fold partition diamonds, which have counting function <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. Using partition analysis, we then find the generating function for both, and connect the generating functions <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan–like congruences satisfied by <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for various values of <em>d</em>, including the following family: for all <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span> and all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"d-fold partition diamonds\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114163\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this work we introduce new combinatorial objects called <em>d</em>–fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> to be their counting function. We also consider the Schmidt type <em>d</em>–fold partition diamonds, which have counting function <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. Using partition analysis, we then find the generating function for both, and connect the generating functions <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan–like congruences satisfied by <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for various values of <em>d</em>, including the following family: for all <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span> and all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24002942\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24002942","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
摘要
在这项工作中,我们引入了新的组合对象,称为 d 折分菱形,它既概括了经典的分区函数,也概括了安德鲁斯、波尔和里埃斯的分区菱形,并将 rd(n) 设为它们的计数函数。我们还考虑了具有计数函数 sd(n) 的施密特型 d 折分菱形。利用分区分析,我们可以找到两者的生成函数,并将生成函数∑n=0∞sd(n)qn 与欧拉多项式联系起来。这样,我们就能为 sd(n) 在不同 d 值下满足的无穷多个类似拉马努扬的同余式进行基本证明,包括以下系列:对于所有 d≥1 和所有 n≥0,sd(2n+1)≡0(mod2d)。
In this work we introduce new combinatorial objects called d–fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set to be their counting function. We also consider the Schmidt type d–fold partition diamonds, which have counting function . Using partition analysis, we then find the generating function for both, and connect the generating functions to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan–like congruences satisfied by for various values of d, including the following family: for all and all , .
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.