Journal of Combinatorial Theory Series A最新文献

英文中文

Complete 3-term arithmetic progression free sets of small size in vector spaces and other abelian groups 向量空间和其他阿贝尔群中的完全3项等差数列自由小集

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-01 Epub Date: 2025-04-30 DOI: 10.1016/j.jcta.2025.106061

Bence Csajbók , Zoltán Lóránt Nagy

A subset S of an abelian group G is called 3-AP free if it does not contain a three term arithmetic progression. Moreover, S is called complete 3-AP free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a 3-AP free set, which is necessarily complete. In this paper we are interested in the minimum size of complete 3-AP free sets. We define and study saturation w.r.t. 3-APs and present constructions of small complete 3-AP free sets and 3-AP saturating sets for several families of vector spaces and cyclic groups.

如果阿贝尔群G的子集S不包含三个等差数列，则称为3-AP自由子集S。另外，如果S是最大的w.r.t.集合包含，则称为完全3-AP自由。加性组合学中最核心的问题之一是确定一个3-AP自由集合的最大大小，该集合必须是完全的。本文主要研究3-AP完全自由集的最小尺寸问题。我们定义并研究了3-AP的饱和，给出了若干向量空间族和循环群的小完全3-AP自由集和3-AP饱和集的构造。

引用次数: 0

Normal edge-transitive Cayley graphs on non-abelian simple groups 非阿贝尔单群上的正规边传递Cayley图

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-01 Epub Date: 2025-03-31 DOI: 10.1016/j.jcta.2025.106050

Xing Zhang, Yan-Quan Feng, Fu-Gang Yin, Jin-Xin Zhou

Let Γ be a Cayley graph on a finite group G, and let

N_{Aut (Γ)} (R (G))

be the normalizer of

R (G)

(the right regular representation of G) in the full automorphism group

Aut (Γ)

of Γ. We say that Γ is a normal Cayley graph on G if

N_{Aut (Γ)} (R (G)) = Aut (Γ)

, and that Γ is a normal edge-transitive Cayley graph on G if

N_{Aut (Γ)} (R (G))

acts transitively on the edge set of Γ. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency p is normal for each prime p. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if G is either

{PSL}_{2} (q)

for an odd prime

q \geq 5

, or

A_{n}

for

n \geq 5

, then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of G.

设Γ是有限群G上的Cayley图，设NAut（Γ）（R(G)）是Γ的完全自同构群Aut（Γ）中R(G) （G的右正则表示）的归一化器。我们说，如果NAut（Γ）(R(G))=Aut（Γ）， Γ是G上的正规Cayley图；如果NAut（Γ）（R(G)）传递作用于Γ的边集，Γ是G上的正规边传递Cayley图。Praeger在1999年证明了在价为3的有限非阿贝简单群上的每一个连通正规边传递Cayley图都是正规的。作为这一结论的推广，我们证明了在价为p的有限非阿贝简单群上，每一个连通正规边传递Cayley图对于每一个素数p都是正规的，而对于合价则不成立。给出了构造若干群的连通正规边可传递非正规Cayley图的一种方法，并由此证明了当奇数素数q≥5时G为PSL2(q)，或n≥5时G为an，则存在G的连通正规边可传递非正规8价Cayley图。

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引用次数: 0

Finite versions of the Andrews–Gordon identity and Bressoud's identity 安德鲁斯-戈登恒等式和布雷苏德恒等式的有限版本

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-01 Epub Date: 2025-03-18 DOI: 10.1016/j.jcta.2025.106035

Heng Huat Chan , Song Heng Chan

In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.

在这篇文章中，我们讨论了欧拉五边形数特性的有限版本、罗杰斯-拉玛努扬特性，并提出了安德鲁斯-戈登特性和布里苏德特性有限版本的新证明。我们还研究了加尔文对戴森秩的泛化的有限版本，并发现了安德鲁斯-戈登同一性的一个新的单变量扩展。

引用次数: 0

Characterization of polystochastic matrices of order 4 with zero permanent 零永久的4阶多随机矩阵的表征

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-01 Epub Date: 2025-04-30 DOI: 10.1016/j.jcta.2025.106060

Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if d is even, then the permanent of a d-dimensional polystochastic matrix of order 4 is positive, and for odd d, we give a complete characterization of d-dimensional polystochastic matrices with zero permanent.

一个多维非负矩阵，如果其每一行上的元素之和等于1，则称为多随机矩阵。多维矩阵的恒量是所有对角线上元素的乘积的和。证明了如果d是偶数，则4阶的d维多随机矩阵的永久性是正的，对于奇数d，我们给出了零永久性的d维多随机矩阵的完整刻画。

引用次数: 0

Studying the divisibility of power LCM matrices by power GCD matrices on gcd-closed sets 研究了幂LCM矩阵在GCD闭集上被幂GCD矩阵可整除的问题

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-01 Epub Date: 2025-05-02 DOI: 10.1016/j.jcta.2025.106063

Jianrong Zhao , Chenxu Wang , Yu Fu

<div><div>Let <math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math> be a gcd-closed set (i.e. <math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>S</mi></math> for all <math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math>). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets S with <math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≥</mo><mn>4</mn></math> such that the GCD matrix (S) divides the LCM matrix <math><mo>[</mo><mi>S</mi><mo>]</mo></math> in the ring <math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math>. For <math><mi>x</mi><mo>∈</mo><mi>S</mi></math>, let <math><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>S</mi><mo>:</mo><mi>z</mi><mo><</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>|</mo><mi>x</mi><mtext> and </mtext><mo>(</mo><mi>z</mi><mo>|</mo><mi>y</mi><mo>|</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>)</mo><mo>⇒</mo><mi>y</mi><mo>∈</mo><mo>{</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>}</mo><mo>}</mo></math>. In 2009, Feng, Hong and Zhao answered this problem in the context where <math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo>⁡</mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≤</mo><mn>2</mn></math>. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set S with <math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo>⁡</mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>=</mo><mn>3</mn></math> such that <math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math>. Meanwhile, they raised a conjecture on the necessary and sufficient condition such that <math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math> holds for the remaining case <math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo>⁡</mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≥</mo><mn>4</mn></math>. In this paper, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.</d

设S={x1，…，xn}是一个gcd闭集（即(xi,xj)∈S，对于所有1≤i，j≤n）。2002年，Hong提出了表征所有GCD -闭集S的|S|≥4使得GCD矩阵(S)能除环Mn(Z)中的LCM矩阵[S]的可分性问题。x∈年代,让GS (x): = {z∈年代:z< x, z | x和y z | | x, y∈(S)⇒y∈{z、x}}。2009年，Feng， Hong和Zhao在maxx∈S∈{|GS(x)|}≤2的情况下回答了这个问题。Zhao、Chen和Hong在2022年得到了maxx∈S∈S (|GS(x)|}=3的gcd-闭集S上的一个充要条件，使得(S)|[S]。同时，他们提出了一个关于(S)|[S]对剩余情况maxx∈S∈{|GS(x)|}≥4成立的充分必要条件的猜想。本文从一个全新的角度证实了赵-陈-洪猜想，从而彻底解决了洪的开放问题。

引用次数: 0

The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups 对称群上一些非正态Cayley图的第二大特征值

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-08 DOI: 10.1016/j.jcta.2025.106097

Yuxuan Li, Binzhou Xia, Sanming Zhou

A Cayley graph on the symmetric group <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For <mml:math altimg="si2.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>n</mml:mi></mml:math>, let <mml:math altimg="si267.svg"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> be the set of <ce:italic>k</ce:italic>-cycles of <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> moving every point in <mml:math altimg="si4.svg"><mml:mo stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:math>. Recently, Siemons and Zalesski (2022) <ce:cross-ref ref>[26]</ce:cross-ref> posed a conjecture which is equivalent to saying that for any <mml:math altimg="si5.svg"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>5</mml:mn></mml:math> and <mml:math altimg="si2.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>n</mml:mi></mml:math> the nonnormal Cayley graph <mml:math altimg="si6.svg"><mml:mrow><mml:mi mathvariant="normal">Cay</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> on <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> with connection set <mml:math altimg="si267.svg"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) <mml:math altimg="si7.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi

如果对称群Sn上的Cayley图的严格第二大特征值（即严格小于度的最大特征值）通过Sn的标准表示获得，则称其具有Aldous性质。对于1≤r<；k<n，设C（n,k;r）为Sn移动{1，…，r}中每一点的k个环的集合。最近，Siemons and Zalesski(2022)[26]提出了一个猜想，该猜想等价于对于任意n≥5且1≤r<；k<n，具有连接集C（n,k;r）的Sn上的非正态Cayley图Cay（Sn，C(n,k;r)）具有Aldous性质。通过求解这个猜想，我们证明了除(i) (n,k,r)=(6,5,1)或（ii） n为奇数，k =n−1，且1≤r<；n2外，所有图都具有Aldous性质。在此过程中，我们确定了Sn的所有不可约表示，这些表示可以实现Cay（Sn，C(n,n - 1;r)）的严格第二大特征值以及该图的最小特征值。

引用次数: 0

The geometry of intersecting codes and applications to additive combinatorics and factorization theory 交码几何及其在加性组合学和分解理论中的应用

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-01 Epub Date: 2025-02-27 DOI: 10.1016/j.jcta.2025.106023

Martino Borello , Wolfgang Schmid , Martin Scotti

Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicities) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted 2-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.

交码是线性码，其中每两个非零码字都有非平凡的交支持。在本文中，我们通过证明非退化相交码对应于射影空间中不包含在两个超平面中的点集（具有多重性）来扩展这类码的理论。这种对应关系允许使用几何参数来演示属性并提供相交代码的构造。我们改进了现有的边界长度，并提供了短相交码的显式结构。最后，推广编码理论与有限阿贝尔群的组合不变量Davenport常数理论之间的联系，给出了加权2-wise Davenport常数的新的渐近界。然后，这些界给出了代数整数环和相关结构的因数分解的结果。

引用次数: 0

A central limit theorem for a card shuffling problem 洗牌问题的中心极限定理

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-01 Epub Date: 2025-04-03 DOI: 10.1016/j.jcta.2025.106048

Shane Chern , Lin Jiu , Italo Simonelli

Given a positive integer n, consider a permutation of n objects chosen uniformly at random. In this permutation, we collect maximal subsequences consisting of consecutive numbers arranged in ascending order called blocks. Each block is then merged, and after all merges, the elements of this new set are relabeled from 1 to the current number of elements. We continue to permute and merge this new set uniformly at random until only one object is left. In this paper, we investigate the distribution of

X_{n}

, the number of permutations needed for this process to end. In particular, we find explicit asymptotic expressions for the mean value

E [X_{n}]

, the variance

Var [X_{n}]

, and higher central moments, and show that

X_{n}

satisfies a central limit theorem.

给定一个正整数n，考虑随机均匀选择的n个对象的排列。在这种排列中，我们收集由按升序排列的连续数字组成的最大子序列，称为块。然后合并每个块，在所有合并之后，这个新集合的元素被重新标记，从1到当前的元素数。我们继续均匀随机地排列和合并这个新集合，直到只剩下一个对象。在本文中，我们研究了Xn的分布，即该过程结束所需的排列数。特别地，我们找到了均值E[Xn]、方差Var[Xn]和高中心矩的显式渐近表达式，并证明了Xn满足中心极限定理。

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引用次数: 0

Separable elements and splittings in Weyl groups of type B B型Weyl群的可分离元素与分裂

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-01 Epub Date: 2025-02-27 DOI: 10.1016/j.jcta.2025.106021

Ming Liu, Houyi Yu

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair

(X, Y)

of subsets of the symmetric group

S_{n}

, the multiplication map

X \times Y \to S_{n}

is a splitting (i.e., a length-additive bijection) of

S_{n}

if and only if X is the generalized quotient of Y and Y is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type B.

Weyl群中的可分离元素是对称群中著名的可分离置换的推广。Gaetz和Gao证明了对于对称群Sn的任意子集对（X，Y），当且仅当X是Y的广义商，Y是由可分离元素生成的右弱阶主低阶理想时，乘法映射X×Y→Sn是Sn的分裂（即长度加性双射）。他们推测这个结果可以推广到所有有限Weyl群。本文用禁止模式对所有可分和最小不可分有符号置换进行了分类，并证实了B型Weyl群的Gaetz和Gao猜想。

引用次数: 0

Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances 满足Griesmer界或具有最优最小距离的二进制自正交码

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-01 Epub Date: 2025-02-28 DOI: 10.1016/j.jcta.2025.106027

Minjia Shi , Shitao Li , Tor Helleseth , Jon-Lark Kim

The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing the Solomon-Stiffler codes. As a result, we reduce a problem with an infinite number of cases to a finite number of cases. Second, we develop a general method to prove the nonexistence of some binary self-orthogonal codes by considering the residual code of a binary self-orthogonal code. Using such a characterization, we completely determine the exact value of

d_{s o} (n, 7)

, where

d_{s o} (n, k)

denotes the largest minimum distance among all binary self-orthogonal

[n, k]

codes.

本文的目的是双重的。首先，我们利用solomon - stiff码刻画了满足Griesmer界的二进制自正交码的存在性。因此，我们将一个有无限种情况的问题简化为有限种情况。其次，通过考虑二进制自正交码的残差码，给出了证明某些二进制自正交码不存在的一般方法。利用这样的表征，我们完全确定了dso（n,7）的精确值，其中dso（n,k）表示所有二进制自正交[n，k]码之间的最大最小距离。

引用次数: 0

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