{"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}
引用次数: 0
Abstract
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.