{"title":"Arithmetic Properties of Certain t-Regular Partitions","authors":"Rupam Barman, Ajit Singh, Gurinder Singh","doi":"10.1007/s00026-023-00649-z","DOIUrl":null,"url":null,"abstract":"<div><p>For a positive integer <span>\\(t\\ge 2\\)</span>, let <span>\\(b_{t}(n)\\)</span> denote the number of <i>t</i>-regular partitions of a nonnegative integer <i>n</i>. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for <span>\\(b_9(n)\\)</span> and <span>\\(b_{19}(n)\\)</span>. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of <span>\\(b_9(n)\\)</span> and <span>\\(b_{19}(n)\\)</span> modulo 2. For <span>\\(t\\in \\{6,10,14,15,18,20,22,26,27,28\\}\\)</span>, Keith and Zanello conjectured that there are no integers <span>\\(A>0\\)</span> and <span>\\(B\\ge 0\\)</span> for which <span>\\(b_t(An+ B)\\equiv 0\\pmod 2\\)</span> for all <span>\\(n\\ge 0\\)</span>. We prove that, for any <span>\\(t\\ge 2\\)</span> and prime <span>\\(\\ell \\)</span>, there are infinitely many arithmetic progressions <span>\\(An+B\\)</span> for which <span>\\(\\sum _{n=0}^{\\infty }b_t(An+B)q^n\\not \\equiv 0 \\pmod {\\ell }\\)</span>. Next, we obtain quantitative estimates for the distributions of <span>\\(b_{6}(n), b_{10}(n)\\)</span> and <span>\\(b_{14}(n)\\)</span> modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and 13-regular partition functions.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 2","pages":"439 - 457"},"PeriodicalIF":0.6000,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00649-z","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
For a positive integer \(t\ge 2\), let \(b_{t}(n)\) denote the number of t-regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for \(b_9(n)\) and \(b_{19}(n)\). We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of \(b_9(n)\) and \(b_{19}(n)\) modulo 2. For \(t\in \{6,10,14,15,18,20,22,26,27,28\}\), Keith and Zanello conjectured that there are no integers \(A>0\) and \(B\ge 0\) for which \(b_t(An+ B)\equiv 0\pmod 2\) for all \(n\ge 0\). We prove that, for any \(t\ge 2\) and prime \(\ell \), there are infinitely many arithmetic progressions \(An+B\) for which \(\sum _{n=0}^{\infty }b_t(An+B)q^n\not \equiv 0 \pmod {\ell }\). Next, we obtain quantitative estimates for the distributions of \(b_{6}(n), b_{10}(n)\) and \(b_{14}(n)\) modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and 13-regular partition functions.
期刊介绍:
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches