Arithmetic Properties of Certain t-Regular Partitions

IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED
Rupam Barman, Ajit Singh, Gurinder Singh
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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.

某些t-正则分区的算术性质
对于一个正整数\(t\ge 2\), 让\(b_{t}(n)\)表示一个非负整数 n 的 t-regular partitions 的个数。受 Keith 和 Zanello 最近的一些猜想的启发,我们为\(b_9(n)\) 和\(b_{19}(n)\)建立了 modulo 2 的无限全等族。我们证明了 Keith 和 Zanello 关于 \(b_9(n)\) 和 \(b_{19}(n)\) modulo 2 的自相似性的两个猜想的一些具体情况。对于 \(t\in {6,10,14,15,18,20,22,26,27,28}/),基思和扎内罗猜想,对于所有的 \(n\ge 0\) ,不存在整数 \(A>0\) 和 \(B\ge 0\) 。我们证明,对于任意的(t\ge 2\ )和素数(ell\),有无限多的算术级数(\(sum _{n=0}^{\infty }b_t(An+B)q^not (equiv 0 (pmod {\ell }\ ))。接下来,我们得到了 \(b_{6}(n), b_{10}(n)\) 和 \(b_{14}(n)\) modulo 2 分布的定量估计。我们进一步研究了与 7-regular 和 13-regular 分割函数相关的某些无穷等差数列的奇数密度。
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来源期刊
Annals of Combinatorics
Annals of Combinatorics 数学-应用数学
CiteScore
1.00
自引率
0.00%
发文量
56
审稿时长
>12 weeks
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
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