Journal of Combinatorial Theory Series A最新文献

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The enumeration of equivalent classes of minimal general dihedral group codes 最小一般二面群编码等价类的枚举

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-26 DOI: 10.1016/j.jcta.2024.105906

Boheng Huang

A group code is a linear code which can be realized as a two-sided ideal of a group algebra over a finite field. When the characteristic of the field is prime to the order of the group, we will give explicit expressions for primitive central idempotents in the group algebra, which enables us to determine the number of equivalent classes of minimal group codes. Then, we apply our formula to calculate the number of equivalent classes of minimal general dihedral group codes.

群码是一种线性码，可以作为有限域上的群代数的双面理想来实现。当场的特征是群的阶的素数时，我们将给出群代数中原始中心幂的明确表达式，这使我们能够确定最小群码的等价类数。然后，我们将应用我们的公式计算最小一般二面体群码的等价类数。

引用次数: 0

The number of primitive words of unbounded exponent in the language of an HD0L-system is finite 在 HD0L 系统的语言中，无限制指数的基元字的数量是有限的

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-22 DOI: 10.1016/j.jcta.2024.105904

Karel Klouda, Štěpán Starosta

Let H be an HD0L-system. We show that there are only finitely many primitive words v with the property that $v^{k}$ , for all integers k, is an element of the factorial language of H. In particular, this result applies to the set of all factors of a morphic word. We provide a formalized proof in the proof assistant Isabelle/HOL as part of the Combinatorics on Words Formalized project.

设 H 是一个 HD0L 系统。我们证明，对于所有整数 k，只有有限多个原始词 v 具有这样的性质：vk 是 H 的因子语言的一个元素。我们在证明助手 Isabelle/HOL 中提供了形式化证明，这是词上组合论形式化项目的一部分。

引用次数: 0

Basic tetravalent oriented graphs with cyclic normal quotients 具有循环法商的基本四价定向图

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-17 DOI: 10.1016/j.jcta.2024.105895

Nemanja Poznanović , Cheryl E. Praeger

Let $OG (4)$ denote the family of all graph-group pairs $(Γ, G)$ where Γ is finite, 4-valent, connected, and G-oriented (G-half-arc-transitive). A subfamily of $OG (4)$ has recently been identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one basic member. In this paper we provide a description of such basic pairs which have at least one G-normal quotient which is isomorphic to a cycle graph. In doing so, we produce many new infinite families of examples and solve several problems posed in the recent literature on this topic. This result completes a research project aiming to provide a description of all basic pairs in $OG (4)$ .

让 OG(4) 表示所有图-群对（Γ,G）的族，其中Γ 是有限的、四价的、连通的和面向 G 的（G-半弧-传递性）。最近，OG(4)的一个子族被认定为 "基本 "族，因为该族中的所有图都是至少一个基本成员的法向盖。在本文中，我们描述了至少有一个 G 常商数与循环图同构的基本图对。在此过程中，我们产生了许多新的无穷族实例，并解决了最近有关这一主题的文献中提出的几个问题。这一成果完成了一个旨在描述 OG(4) 中所有基本对的研究项目。

引用次数: 0

Explicit formulas for a family of hypermaps beyond the one-face case 超越单面情况的超映射族的明确公式

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-17 DOI: 10.1016/j.jcta.2024.105905

Zi-Wei Bai, Ricky X.F. Chen

Enumeration of hypermaps (or Grothendieck's dessins d'enfants) is widely studied in many fields. In particular, enumerating hypermaps with a fixed edge-type according to the number of faces and genus is one topic of great interest. The first systematic study of hypermaps with one face and any fixed edge-type is the work of Jackson (1987) [23] using group characters. Stanley later (2011) obtained the genus distribution polynomial of one-face hypermaps of any fixed edge-type expressed in terms of the backward shift operator. There is also enormous amount of work on enumerating one-face hypermaps of specific edge-types. Hypermaps with more faces are generally much harder to enumerate and results are rare. Our main results here are formulas for the genus distribution polynomials for a family of typical two-face hypermaps including almost all edge-types, the purely imaginary zeros property of these polynomials, and the log-concavity of the coefficients.

超映射的枚举（或格罗登第克的儿童图）在许多领域被广泛研究。其中，根据面数和属数枚举具有固定边型的超映射是一个备受关注的课题。杰克逊（Jackson，1987 年）[23] 利用群特征首次系统地研究了具有一个面和任意固定边型的超映射。Stanley 后来（2011 年）获得了用后移算子表示的任意固定边型的单面超映射的属分布多项式。在枚举特定边类型的单面超映射方面也有大量工作。面数更多的超映射通常更难枚举，结果也很少见。我们在这里的主要成果是一系列典型两面超映射（包括几乎所有边缘类型）的属分布多项式公式、这些多项式的纯虚零属性以及系数的对数凹性。

引用次数: 0

Proof of a conjecture of Ballantine and Merca on truncated sums of 6-regular partitions 巴兰坦和梅尔卡关于 6 不规则分区截断和的猜想的证明

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-15 DOI: 10.1016/j.jcta.2024.105903

Olivia X.M. Yao

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Their work has opened up a new study of truncated series. Recently, Ballantine and Merca posed a conjecture on infinite families of inequalities involving $b_{6} (n)$ , which counts the number of 6-regular partitions of n. In this paper, we confirm Ballantine and Merca's conjecture on linear equalities of $b_{6} (n)$ based on a formula of the number of partitions of n into parts not exceeding 3 due to Cayley.

2012 年，安德鲁斯和梅尔卡证明了欧拉五边形数定理的截断定理。他们的工作开启了截断数列的新研究。最近，Ballantine 和 Merca 提出了一个关于涉及 b6(n) 的无限不等式族的猜想，该猜想计算了 n 的 6 规则分区数。在本文中，我们根据 Cayley 提出的将 n 分成不超过 3 部分的公式，证实了 Ballantine 和 Merca 关于 b6(n) 线性等式的猜想。

引用次数: 0

Classical groups as flag-transitive automorphism groups of 2-designs with λ = 2 经典群作为 λ = 2 的 2 设计的旗跨自变群

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-12 DOI: 10.1016/j.jcta.2024.105892

Seyed Hassan Alavi , Mohsen Bayat , Ashraf Daneshkhah , Marjan Tadbirinia

In this article, we study 2-designs with $λ = 2$ admitting a flag-transitive and point-primitive almost simple automorphism group G with socle X a finite simple classical group of Lie type. We prove that such a design belongs to an infinite family of 2-designs with parameter set $((3^{n} - 1) / 2, 3, 2)$ and $X = {PSL}_{n} (3)$ for some $n ⩾ 3$ , or $X = {PSL}_{2} (q)$ with point-stabiliser $D_{2 (q + 1) / \gcd (2, q - 1)}$ , or it is isomorphic to the 2-design with parameter set $(6, 3, 2)$ , $(7, 4, 2)$ , $(10, 4, 2)$ , $(11, 5, 2)$ , $(28, 7, 2)$ , $(28, 3, 2)$ , $(36, 6, 2)$ or $(126, 6, 2)$ .

在本文中，我们研究了 λ=2 的 2 设计，它容许一个旗递和点直立的几乎简单的自动形群 G，其共面 X 是一个有限简单的李型经典群。我们证明，这样的设计属于参数集为（(3n-1)/2,3,2）且 X=PSLn(3) 对于某个 n⩾3，或 X=PSL2(q) 具有点稳定器 D2(q+1)/gcd(2、q-1），或与参数集为（6,3,2）、（7,4,2）、（10,4,2）、（11,5,2）、（28,7,2）、（28,3,2）、（36,6,2）或（126,6,2）的 2 设计同构。

引用次数: 0

Further q-reflections on the modulo 9 Kanade–Russell (conjectural) identities 对模数 9 卡纳德-罗素（猜想）特性的进一步 q- 反思

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-10 DOI: 10.1016/j.jcta.2024.105894

Stepan Konenkov

We examine four identities conjectured by Dean Hickerson which complement five modulo 9 Kanade–Russell identities, and we build up a profile of new identities and new conjectures similar to those found by Ali Uncu and Wadim Zudilin.

我们研究了迪安-希克森（Dean Hickerson）猜想的四个特性，这些特性补充了五个模态 9 卡纳德-罗素特性，我们还建立了一个新特性和新猜想的概况，与阿里-温库（Ali Uncu）和瓦迪姆-祖迪林（Wadim Zudilin）发现的特性和猜想相似。

引用次数: 0

A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids 欧拉分支贪婪多项式根无关性的几何证明

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-04 DOI: 10.1016/j.jcta.2024.105891

Lilla Tóthmérész

We define the root polytope of a regular oriented matroid, and show that the greedoid polynomial of an Eulerian branching greedoid rooted at vertex $v_{0}$ is equivalent to the $h^{⁎}$ -polynomial of the root polytope of the dual of the graphic matroid.

As the definition of the root polytope is independent of the vertex $v_{0}$ , this gives a geometric proof for the root-independence of the greedoid polynomial for Eulerian branching greedoids, a fact which was first proved by Swee Hong Chan, Kévin Perrot and Trung Van Pham using sandpile models. We also obtain that the greedoid polynomial does not change if we reverse every edge of an Eulerian digraph.

我们定义了正则定向 matroid 的根多胞形，并证明了以顶点 v0 为根的欧拉分支 greedoid 的 greedoid 多项式等价于图形 matroid 对偶的根多胞形的⁎-多项式。由于根多胞形的定义与顶点 v0 无关，这就给出了欧拉分支 greedoid 多项式与根无关的几何证明，而这一事实最早是由 Swee Hong Chan、Kévin Perrot 和 Trung Van Pham 利用沙堆模型证明的。我们还得出，如果我们将欧拉图的每条边反转，greedoid 多项式也不会发生变化。

引用次数: 0

Cyclic relative difference families with block size four and their applications 块大小为四的循环相对差分系列及其应用

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-03 DOI: 10.1016/j.jcta.2024.105890

Chenya Zhao , Binwei Zhao , Yanxun Chang , Tao Feng , Xiaomiao Wang , Menglong Zhang

<div>Given a subgroup H of a group <math><mo>(</mo><mi>G</mi><mo>,</mo><mo>+</mo><mo>)</mo></math>, a <math><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math> difference family (DF) is a set <math><mi>F</mi></math> of k-subsets of G such that <math><mo>{</mo><mi>f</mi><mo>−</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>f</mi><mo>,</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>F</mi><mo>,</mo><mi>f</mi><mo>≠</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><mi>F</mi><mo>∈</mo><mi>F</mi><mo>}</mo><mo>=</mo><mi>G</mi><mo>∖</mo><mi>H</mi></math>. Let <math><mi>g</mi><msub><mrow><mi>Z</mi></mrow><mrow><mi>g</mi><mi>h</mi></mrow></msub></math> be the subgroup of order h in <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>g</mi><mi>h</mi></mrow></msub></math> generated by g. A <math><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>g</mi><mi>h</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>Z</mi></mrow><mrow><mi>g</mi><mi>h</mi></mrow></msub><mo>,</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math>-DF is called cyclic and written as a <math><mo>(</mo><mi>g</mi><mi>h</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math>-CDF. This paper shows that for <math><mi>h</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>}</mo></math>, there exists a <math><mo>(</mo><mi>g</mi><mi>h</mi><mo>,</mo><mi>h</mi><mo>,</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>-CDF if and only if <math><mi>g</mi><mi>h</mi><mo>≡</mo><mi>h</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>12</mn><mo>)</mo></math>, <math><mi>g</mi><mo>⩾</mo><mn>4</mn></math> and <math><mo>(</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo>)</mo><mo>∉</mo><mo>{</mo><mo>(</mo><mn>9</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>}</mo></math>. As a corollary, it is shown that a 1-rotational Steiner system S<math><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi>v</mi><mo>)</mo></math> exists if and only if <math><mi>v</mi><mo>≡</mo><mn>4</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>12</mn><mo>)</mo></math> and <math><mi>v</mi><mo>≠</mo><mn>28</mn></math>. This solves the long-standing open problem on the existence of a 1-rotational S<math><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi>v</mi><mo>)</mo></math>. As another corollary, we establish the existence of an optimal <math><mo>(</mo><mi>v</mi><mo>,</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>-optical orthogonal code with <math><mo>⌊</mo><mo>(</mo><mi>v</mi><mo>−</

给定一个群（G,+）的子群 H，一个（G,H,k,1）差族（DF）是 G 的 k 个子集的集合 F，使得 {f-f′:f,f′∈F,f≠f′,F∈F}=G∖H。设 gZgh 是 g 在 Zgh 中产生的阶为 h 的子群。(Zgh,gZgh,k,1)-DF 称为循环DF，并写成 (gh,h,k,1)-CDF。本文指出，对于 h∈{2,3,6}，当且仅当 gh≡h（mod12），g⩾4 且 (g,h)∉{(9,3),(5,6)} 时，存在一个 (gh,h,4,1)-CDF 。推论表明，当且仅当 v≡4（mod12）且 v≠28 时，存在一个 1 旋转的斯坦纳系统 S(2,4,v)。这就解决了存在 1- 旋转 S(2,4,v) 这一长期悬而未决的问题。作为另一个推论，我们确定了对于任意正整数 v≡1,2,3,4,6(mod12)和 v≠25，存在一个具有⌊(v-1)/12⌋码字的最优 (v,4,1) 光正交码。我们还给出了我们的结果在块大小为四的循环群可分设计和权重为四且最小距离为六的最优循环三元恒权码中的应用。

引用次数: 0

The complex genera, symmetric functions and multiple zeta values 复属、对称函数和多重zeta值

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-04-03 DOI: 10.1016/j.jcta.2024.105893

Ping Li

We examine the coefficients in front of Chern numbers for complex genera, and pay special attention to the ${Td}^{\frac{1}{2}}$ -genus, the Γ-genus as well as the Todd genus. Some related geometric applications to hyper-Kähler and Calabi-Yau manifolds are discussed. Along this line and building on the work of Doubilet in 1970s, various Hoffman-type formulas for multiple-(star) zeta values and transition matrices among canonical bases of the ring of symmetric functions can be uniformly treated in a more general framework.

我们研究了复属的切尔数前面的系数，并特别关注 Td12 属、Γ 属和 Todd 属。还讨论了超凯勒流形和卡拉比-尤流形的一些相关几何应用。沿着这一思路，在杜比莱 1970 年代工作的基础上，各种霍夫曼式的多重（星形）zeta 值公式和对称函数环的典范基之间的过渡矩阵可以在一个更普遍的框架内统一处理。

引用次数: 0

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