Journal of Combinatorial Theory Series A最新文献

英文中文

Two conjectures of Andrews, Merca and Yee on truncated theta series 安德鲁斯、梅尔卡和易关于截断θ级数的两个猜想

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-22 DOI: 10.1016/j.jcta.2024.105874

Shane Chern , Ernest X.W. Xia

In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function $M_{k} (n)$ to count the number of partitions of n in which k is the least integer that is not a part and there are more parts exceeding k than there are below k. In recent years, two conjectures concerning $M_{k} (n)$ on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large n whenever k is fixed.

安德鲁斯和梅尔卡在研究欧拉五边形数截断定理时，引入了一个分区函数 Mk(n)，用来计算 n 的分区数，其中 k 是不属于分区的最小整数，且超过 k 的分区数多于低于 k 的分区数。近年来，安德鲁斯、梅尔卡和易提出了关于截断θ数列 Mk(n) 的两个猜想。在本文中，我们证明了只要 k 固定不变，对于足够大的 n，这两个猜想都是真的。

引用次数: 0

Constructing generalized Heffter arrays via near alternating sign matrices 通过近交替符号矩阵构建广义赫夫特阵列

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-21 DOI: 10.1016/j.jcta.2024.105873

L. Mella , T. Traetta

<div>Let S be a subset of a group G (not necessarily abelian) such that <math><mi>S</mi><mspace></mspace><mo>∩</mo><mo>−</mo><mi>S</mi></math> is empty or contains only elements of order 2, and let <math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>m</mi></mrow></msup></math> and <math><mi>k</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. A generalized Heffter array GHA<math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> over G is an <math><mi>m</mi><mo>×</mo><mi>n</mi></math> matrix <math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math> such that: the i-th row (resp. j-th column) of A contains exactly <math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math> (resp. <math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math>) nonzero elements, and the list <math><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>}</mo></math> equals λ times the set <math><mi>S</mi><mspace></mspace><mo>∪</mo><mspace></mspace><mo>−</mo><mi>S</mi></math>. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of A sums to zero (resp. a nonzero element), with respect to some ordering.In this paper, we use near alternating sign matrices to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to 0 modulo 4. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular (<math><mi>m</mi><mo>≠</mo><mi>n</mi></math>) and with less than n nonzero entries in each row. Furthermore, we build nonzero sum GHA<math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> over an arbitrary group G whenever S contains enough noninvolutions, th

设 S 是一个群 G 的子集（不一定是非良性的），使得 S∩-S 是空的或只包含阶数为 2 的元素，并且设 h=（h1,...,hm）∈Nm，k=（k1,...,kn ）∈Nn。G 上的广义赫夫特数组 GHASλ(m,n;h,k) 是一个 m×n 矩阵 A=(aij)，使得：A 的第 i 行（或第 j 列）恰好包含 hi（或 kj）个非零元素，且列表 {aij,-aij|aij≠0} 等于集合 S∪-S 的 λ 倍。在本文中，我们使用近交替符号矩阵来构建循环群上的零和（或非零和）GHA，它还具有简单的强性质。特别是，我们构建的零和简单 GHA，其行权重和列权重同余为 0 modulo 4。这一结果还提供了第一个矩形（m≠n）且每行非零条目少于 n 个的简单（经典）赫夫特数组无穷族。此外，只要 S 包含足够多的非卷积，我们就能在任意群 G 上建立非零和 GHASλ(m,n;h,k)，从而扩展了之前的非构造性结果，即对于 G 的某个子群 H，±S=G∖H。最后，我们描述了如何利用 GHA 来建立 Cayley 图（在不一定是无性的群上）到可定向曲面上的正交分解和双嵌套。

引用次数: 0

On the maximal number of elements pairwise generating the finite alternating group 关于成对生成有限交替群的元素的最大数目

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-14 DOI: 10.1016/j.jcta.2024.105870

Francesco Fumagalli , Martino Garonzi , Pietro Gheri

Let G be the alternating group of degree n. Let $ω (G)$ be the maximal size of a subset S of G such that $〈 x, y 〉 = G$ whenever $x, y \in S$ and $x \neq y$ and let $σ (G)$ be the minimal size of a family of proper subgroups of G whose union is G. We prove that, when n varies in the family of composite numbers, $σ (G) / ω (G)$ tends to 1 as $n \to \infty$ . Moreover, we explicitly calculate $σ (A_{n})$ for $n \geq 21$ congruent to 3 modulo 18.

设 G 是 n 阶交替群。设 ω(G) 是 G 的子集 S 的最大大小，当 x,y∈S 且 x≠y 时，使得〈x,y〉=G；设 σ(G) 是 G 的一族适当子群的最小大小，其联合是 G。我们证明，当 n 在合数族中变化时，σ(G)/ω(G) 随着 n→∞ 趋于 1。此外，我们还明确地计算了 n≥21 的 σ(An)与 3 的同余式 18 的同余式。

引用次数: 0

Most plane curves over finite fields are not blocking 有限域上的大多数平面曲线都不是阻塞的

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-09 DOI: 10.1016/j.jcta.2024.105871

Shamil Asgarli , Dragos Ghioca , Chi Hoi Yip

A plane curve $C \subset P^{2}$ of degree d is called blocking if every $F_{q}$ -line in the plane meets C at some $F_{q}$ -point. We prove that the proportion of blocking curves among those of degree d is $o (1)$ when $d \geq 2 q - 1$ and $q \to \infty$ . We also show that the same conclusion holds for smooth curves under the somewhat weaker condition $d \geq 3 p$ and $d, q \to \infty$ . Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of $F_{q}$ -roots of random polynomials, we find that the limiting distribution of the number of $F_{q}$ -points in the intersection of a random plane curve and a fixed $F_{q}$ -line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of $F_{q}$ -points contained in a union of k lines for $k = 1, 2, \dots, q^{2} + q + 1$ .

如果平面中的每条 Fq 线都与 C 交于某个 Fq 点，则阶数为 d 的平面曲线 C⊂P2 称为阻塞曲线。我们证明，当 d≥2q-1 且 q→∞ 时，阻塞曲线在 d 阶曲线中所占比例为 o(1)。我们还证明，在较弱的条件 d≥3p 和 d,q→∞ 下，同样的结论也适用于光滑曲线。此外，随机平面曲线的光滑和阻塞两种情况被证明是渐近独立的。我们扩展了关于随机多项式 Fq 根数的经典结果，发现随机平面曲线与固定 Fq 线交点的 Fq 点数的极限分布是均值为 1 的泊松分布。我们还给出了阻塞曲线比例的明确公式，其中涉及 k=1,2,...q2+q+1 时 k 线结合处所含 Fq 点数的统计量。

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引用次数: 0

A Q-polynomial structure for the Attenuated Space poset Aq(N,M) 衰减空间正集 Aq(N,M) 的 Q 多项式结构

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-09 DOI: 10.1016/j.jcta.2024.105872

Paul Terwilliger

<div>The goal of this article is to display a Q-polynomial structure for the Attenuated Space poset <math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math>. The poset <math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math> is briefly described as follows. Start with an <math><mo>(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>)</mo></math>-dimensional vector space H over a finite field with q elements. Fix an M-dimensional subspace h of H. The vertex set X of <math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math> consists of the subspaces of H that have zero intersection with h. The partial order on X is the inclusion relation. The Q-polynomial structure involves two matrices <math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∈</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math> with the following entries. For <math><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>X</mi></math> the matrix A has <math><mo>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></math>-entry 1 (if y covers z); <math><msup><mrow><mi>q</mi></mrow><mrow><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math> (if z covers y); and 0 (if neither of <math><mi>y</mi><mo>,</mo><mi>z</mi></math> covers the other). The matrix <math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> is diagonal, with <math><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></math>-entry <math><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math> for all <math><mi>y</mi><mo>∈</mo><mi>X</mi></math>. By construction, <math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> has <math><mi>N</mi><mo>+</mo><mn>1</mn></math> eigenspaces. By construction, A acts on these eigenspaces in a (block) tridiagonal fashion. We show that A is diagonalizable, with <math><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn></math> eigenspaces. We show that <math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that A is Q-polynomial. We show that <math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra T of

本文的目的是展示衰减空间正集 Aq(N,M) 的 Q 多项式结构。正集 Aq(N,M) 简述如下。从有限域上具有 q 个元素的 (N+M) 维向量空间 H 开始。Aq(N,M)的顶点集 X 由与 h 有零交集的 H 子空间组成。Q 多项式结构涉及两个矩阵 A,A⁎∈MatX(C)，其条目如下。对于 y,z∈X，矩阵 A 有 (y,z) 项 1（如果 y 覆盖了 z）；qdimy（如果 z 覆盖了 y）；0（如果 y,z 都没有覆盖另一个）。矩阵 A⁎ 是对角线，对于所有 y∈X 都有 (y,y) 项 q-dimy。根据构造，A⁎ 有 N+1 个特征空间。根据构造，A 以（分块）三对角方式作用于这些特征空间。我们证明 A 是可对角的，有 2N+1 个特征空间。我们证明 A⁎ 以（块）三对角方式作用于这些特征空间。利用这一作用，我们证明 A 是 Q 多项式。我们证明 A、A⁎ 满足一对称为三对角关系的关系。我们考虑由 A,A⁎ 生成的 MatX(C) 子代数 T。我们证明，A,A⁎ 作为伦纳德对作用于每个不可还原 T 模块。

引用次数: 0

Spectral characterization of the complete graph removing a cycle 去除一个周期的完整图谱特征

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-09 DOI: 10.1016/j.jcta.2024.105868

Muhuo Liu , Xiaofeng Gu , Haiying Shan , Zoran Stanić

A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph $K_{n} ∖ C_{k}$ , obtained from the complete graph $K_{n}$ with n vertices by deleting all edges of a cycle $C_{k}$ with k vertices, is determined by its spectrum for $k \in {3, 4, 5}$ , but not for $k = 6$ . In this paper, we show that $k = 6$ is the unique exception for the spectral determination of $K_{n} ∖ C_{k}$ .

如果不存在另一个具有相同频谱的图，则该图由其频谱决定。Cámara 和 Haemers 证明，通过删除具有 k 个顶点的循环 Ck 的所有边，从具有 n 个顶点的完整图 Kn 得到的图 Kn∖Ck，在 k∈{3,4,5} 时由其谱决定，但在 k=6 时则不是。在本文中，我们将证明 k=6 是 Kn∖Ck 的谱确定性的唯一例外。

引用次数: 0

The divisor class group of a discrete polymatroid 离散多面体的因子类群

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-08 DOI: 10.1016/j.jcta.2024.105869

Jürgen Herzog , Takayuki Hibi , Somayeh Moradi , Ayesha Asloob Qureshi

In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.

本文介绍了多复数的环。我们展示了如何计算环是正态时的除数类群和典型模块类。在多面体是离散多面体的特殊情况下，我们深入研究了几类多面体的环。

引用次数: 0

Large sum-free sets in Z5n Z5n 中的大无和集

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-02 DOI: 10.1016/j.jcta.2024.105865

Vsevolod F. Lev

It is well-known that for a prime $p \equiv 2 (\mod 3)$ and integer $n \geq 1$ , the maximum possible size of a sum-free subset of the elementary abelian group $Z_{p}^{n}$ is $\frac{1}{3} (p + 1) p^{n - 1}$ . However, the matching stability result is known for $p = 2$ only. We consider the first open case $p = 5$ showing that if $A \subseteq Z_{5}^{n}$ is a sum-free subset with $| A | > \frac{3}{2} \cdot 5^{n - 1}$ , then there are a subgroup $H < Z_{5}^{n}$ of size $| H | = 5^{n - 1}$ and an element $e \notin H$ such that $A \subseteq (e + H) \cup (- e + H)$ .

众所周知，对于素数 p≡2（mod3）和整数 n≥1，初等无方组 Zpn 的无和子集的最大可能大小为 13(p+1)pn-1。然而，已知的匹配稳定性结果仅适用于 p=2。我们考虑第一种开放情况 p=5 表明，如果 A⊆Z5n 是|A|>32⋅5n-1 的无和子集，那么存在大小为 |H|=5n-1的子群 H<Z5n 和元素 e∉H，使得 A⊆(e+H)∪(-e+H)。

引用次数: 0

Block-transitive 2-designs with a chain of imprimitive point-partitions 带有一连串隐含点分区的块变换 2 设计

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-01 DOI: 10.1016/j.jcta.2024.105866

Carmen Amarra , Alice Devillers , Cheryl E. Praeger

More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size k, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of k. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.

30 多年前，德兰切尔和多延证明，大小为 k 块的块变换 2 设计的自动形群可以使一个非难点分区保持不变，但前提是点的数量以 k 为界。自那时起，人们发现了有两个非难点分区的例子，它们要么形成了分区链，要么在点集中形成了网格结构。我们通过构建无穷设计族证明，对于块过渡 2 设计，不变点分区链的长度没有限制。我们引入了点集 "阵列 "的概念，它描述了点集如何与不同分区的部分相互作用，我们还获得了点集 "阵列 "相对于分区链的必要条件和充分条件，从而使其成为这种设计的区块。

引用次数: 0

Large monochromatic components in colorings of complete hypergraphs 完整超图着色中的大型单色成分

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-02-01 DOI: 10.1016/j.jcta.2024.105867

Lyuben Lichev , Sammy Luo

Gyárfás famously showed that in every r-coloring of the edges of the complete graph $K_{n}$ , there is a monochromatic connected component with at least $\frac{n}{r - 1}$ vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for k-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of $1 + o (1)$ as the number of colors grows. We further generalize these questions to ask about counts of vertex s-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.

Gyárfás 的著名研究表明，在完整图 Kn 的边的每 r 种着色中，都存在一个至少有 nr-1 个顶点的单色连通部分。最近，Conlon、Tyomkyn 和第二位作者的一项研究解决了具有许多边的单色连通成分的类似问题。在本文中，我们研究了这些问题在 k-uniform 超图中的推广。在将连通性定义扩展到更高均匀性的广泛范围内，我们为最大单色成分的大小提供了上界和下界，随着颜色数量的增加，上界和下界的紧密程度可达 1+o(1)。我们进一步将这些问题推广到大型单色分量边缘中包含的顶点 s 集的计数。最后，我们将针对两种颜色的特殊情况给出更精确的结果。

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