Journal of Combinatorial Theory Series A最新文献

英文中文

Some conjectures of Ballantine and Merca on truncated sums and the minimal excludant in congruences classes 巴兰廷和梅尔卡关于截断和以及全等类中最小排除因子的一些猜想

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-05 DOI: 10.1016/j.jcta.2024.105967

Olivia X.M. Yao

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Since then, a number of results on truncated theta series have been proved. In this paper, we find the connections between truncated sums of certain partition functions and the minimal excludant statistic which has been found to exhibit connections with a handful of objects such as Dyson's crank. We present a uniform method to confirm five conjectures on truncated sums of certain partition functions given by Ballantine and Merca. In particular, we provide partition-theoretic interpretations for some truncated sums by using the minimal excludant in congruences classes.

2012 年，安德鲁斯和梅尔卡证明了欧拉五角数定理的截断定理。此后，又有许多关于截断θ级数的结果被证明。在本文中，我们发现了某些分割函数的截断和与最小排除统计量之间的联系，而最小排除统计量被发现与戴森曲柄等少数对象存在联系。我们提出了一种统一的方法来证实 Ballantine 和 Merca 提出的关于某些分区函数截断和的五个猜想。特别是，我们通过使用全等类中的最小排除因子，为某些截断和提供了分区理论解释。

引用次数: 0

Reconstruction of hypermatrices from subhypermatrices 从次皮质重建高皮质

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-10-22 DOI: 10.1016/j.jcta.2024.105966

Wenjie Zhong , Xiande Zhang

For a given n, what is the smallest number k such that every sequence of length n is determined by the multiset of all its k-subsequences? This is called the k-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case – reconstruction of

n \times n

-matrices from submatrices. Previous works show that the smallest k is at most

O (n^{\frac{1}{2}})

for sequences and at most

O (n^{\frac{2}{3}})

for matrices. We study this k-deck problem for general dimension d and prove that, the smallest k is at most

O (n^{\frac{d}{d + 1}})

for reconstructing any d dimensional hypermatrix of order n from the multiset of all its subhypermatrices of order k.

对于给定的 n，使得长度为 n 的每个序列都由其所有 k 个子序列的多集决定的最小数 k 是多少？这被称为序列重构的 k 层问题，并已被推广到二维情况--从子矩阵重构 n×n 矩阵。之前的研究表明，对于序列，最小的 k 至多为 O(n12)，而对于矩阵，则至多为 O(n23)。我们研究了一般维数为 d 的 k 层问题，并证明了从其所有阶数为 k 的子超矩阵的多集重构任何阶数为 n 的 d 维超矩阵时，最小 k 至多为 O(ndd+1)。

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引用次数: 0

Direct constructions of column-orthogonal strong orthogonal arrays 列正交强正交阵列的直接构造

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-10-18 DOI: 10.1016/j.jcta.2024.105965

Jingjun Bao , Lijun Ji , Juanjuan Xu

Strong orthogonal arrays have better space-filling properties than ordinary orthogonal arrays for computer experiments. Strong orthogonal arrays of strengths two plus, two star and three minus can improve the space-filling properties in low dimensions and column orthogonality plays a vital role in computer experiments. In this paper, we use difference matrices and generator matrices of linear codes to present several constructions of column-orthogonal strong orthogonal arrays of strengths two plus, two star, three minus and t. Our constructions can provide larger numbers of factors of column-orthogonal strong orthogonal arrays of strengths two plus, two star, three minus and t than those in the existing literature, enjoy flexible run sizes. These constructions are convenient, and the resulting designs are good choices for computer experiments.

在计算机实验中，强正交阵列比普通正交阵列具有更好的空间填充特性。强度为两加、两星和三减的强正交阵列可以改善低维的空间填充特性，而列正交性在计算机实验中起着至关重要的作用。在本文中，我们利用线性编码的差分矩阵和生成矩阵，提出了几种列正交强正交阵列的构造。我们的构造可以提供比现有文献中更多的列正交强正交阵列因子数，并享有灵活的运行规模。这些构造非常方便，由此产生的设计是计算机实验的良好选择。

引用次数: 0

Indecomposable combinatorial games 不可分解的组合博弈

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-10-15 DOI: 10.1016/j.jcta.2024.105964

Michael Fisher , Neil A. McKay , Rebecca Milley , Richard J. Nowakowski , Carlos P. Santos

In Combinatorial Game Theory, short game forms are defined recursively over all the positions the two players are allowed to move to. A form is decomposable if it can be expressed as a disjunctive sum of two forms with smaller birthday. If there are no such summands, then the form is indecomposable. The main contribution of this document is the characterization of the indecomposable nimbers and the characterization of the indecomposable numbers. More precisely, a nimber is indecomposable if and only if its size is a power of two, and a number is indecomposable if and only if its absolute value is less than or equal to one.

在组合博弈论中，简短博弈形式是在允许两位棋手移动的所有位置上递归定义的。如果一个形式可以表示为两个生日较小的形式的析取和，那么这个形式就是可分解的。如果没有这样的和，那么这个棋形就是不可分解的。本文的主要贡献在于描述了不可分解数的特征和不可分解数的特征。更确切地说，当且仅当一个数的大小是 2 的幂时，它是不可分解的；当且仅当一个数的绝对值小于或等于 1 时，它是不可分解的。

引用次数: 0

Point-line geometries related to binary equidistant codes 与二进制等距码有关的点线几何图形

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-10-04 DOI: 10.1016/j.jcta.2024.105962

Mark Pankov , Krzysztof Petelczyc , Mariusz Żynel

Point-line geometries whose singular subspaces correspond to binary equidistant codes are investigated. The main result is a description of automorphisms of these geometries. In some important cases, automorphisms induced by non-monomial linear automorphisms surprisingly arise.

研究了其奇异子空间与二元等距码相对应的点线几何图形。主要结果是描述了这些几何图形的自动形态。在一些重要情况下，非单调线性自动形诱导的自动形出人意料地出现了。

引用次数: 0

Neighborly partitions, hypergraphs and Gordon's identities 邻接分区、超图和戈登等式

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-10-04 DOI: 10.1016/j.jcta.2024.105963

Pooneh Afsharijoo , Hussein Mourtada

We prove a family of partition identities which is “dual” to the family of Andrews-Gordon's identities. These identities are inspired by a correspondence between a special type of partitions and “hypergraphs” and their proof uses combinatorial commutative algebra.

我们证明了与安德鲁斯-戈登（Andrews-Gordon）等价性系列 "对偶 "的分区等价性系列。这些等式的灵感来自于一种特殊类型的分区与 "超图 "之间的对应关系，它们的证明使用了组合交换代数。

引用次数: 0

On locally n × n grid graphs 在局部 n×n 网格图上

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-26 DOI: 10.1016/j.jcta.2024.105957

Carmen Amarra , Wei Jin , Cheryl E. Praeger

We investigate locally

n \times n

grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on n vertices. We consider the subclass of these graphs for which each pair of vertices at distance two is joined by sufficiently many paths of length 2. The number of such paths is known to be at most 2n by previous work of Blokhuis and Brouwer. We show that if each pair is joined by at least

2 (n - 1)

such paths then the diameter is at most 3 and we give a tight upper bound on the order of the graphs. We show that graphs meeting this upper bound are distance-regular antipodal covers of complete graphs. We exhibit an infinite family of such graphs which are locally

n \times n

grid for odd prime powers n, and apply these results to locally

5 \times 5

grid graphs to obtain a classification for the case where either all μ-graphs (induced subgraphs on the set of common neighbours of two vertices at distance two) have order at least 8 or all μ-graphs have order c for some constant c.

我们研究的是局部 n×n 网格图，即任何顶点的邻域都是 n 个顶点上两个完整图的笛卡尔乘积的图。我们考虑的是这些图的子类，其中距离为 2 的每对顶点都有足够多的长度为 2 的路径相连。根据 Blokhuis 和 Brouwer 以前的研究，已知此类路径的数量最多为 2n。我们证明，如果每对图都至少有 2(n-1) 条这样的路径连接，那么直径最多为 3，并且我们给出了图的阶数的严格上限。我们证明，符合这一上限的图是完整图的距离规则反顶盖。我们展示了此类图的一个无穷族，它们在奇素数 n 的情况下是局部 n×n 网格图，并将这些结果应用于局部 5×5 网格图，从而获得了一种分类，即所有 μ 图（距离为 2 的两个顶点的公共相邻集合上的诱导子图）的阶数至少为 8，或者所有 μ 图的阶数为某个常数 c。

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引用次数: 0

On power monoids and their automorphisms 论幂单子及其自动形态

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-25 DOI: 10.1016/j.jcta.2024.105961

Salvatore Tringali, Weihao Yan

Endowed with the binary operation of set addition, the family

P_{fin, 0} (N)

of all finite subsets of

N

containing 0 forms a monoid, with the singleton {0} as its neutral element.

We show that the only non-trivial automorphism of

P_{fin, 0} (N)

is the involution

X \mapsto \max X - X

. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself.

我们证明，Pfin,0(N)的唯一非难自变量是内卷X↦maxX-X。证明利用了加法数论的思想，通过对我们所说的有限整数集合的拳数维度（即其结合为集合本身的最小（离散）区间数）的非常规归纳来进行。

引用次数: 0

On non-empty cross-t-intersecting families 关于非空交叉相交族

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-24 DOI: 10.1016/j.jcta.2024.105960

Anshui Li , Huajun Zhang

Let

A_{1}, A_{2}, \dots, A_{m}

be families of k-element subsets of a n-element set. We call them cross-t-intersecting if

| A_{i} \cap A_{j} | \geq t

for any

A_{i} \in A_{i}

and

A_{j} \in A_{j}

with

i \neq j

. In this paper we will prove that, for

n \geq 2 k - t + 1

, if

A_{1}, A_{2}, \dots, A_{m}

are non-empty cross-t-intersecting families, then

\sum_{1 \leq i \leq m} | A_{i} | \leq \max {(\begin{matrix} n \\ k \end{matrix}) - \sum_{1 \leq i \leq t - 1} (\begin{matrix} k \\ i \end{matrix}) (\begin{matrix} n - k \\ k - i \end{matrix}) + m - 1, m M (n, k, t)},

where

M (n, k, t)

is the size of the maximum t-intersecting family of

(\begin{matrix} [n] \\ k \end{matrix})

. Moreover, the extremal families attaining the upper bound are characterized.

设 A1,A2,...,Am 是 n 元素集合的 k 元素子集族。对于任意 Ai∈Ai 和 Aj∈Aj 且 i≠j 的情况，如果|Ai∩Aj|≥t，我们称它们为交叉-t-交集。本文将证明，对于 n≥2k-t+1，如果 A1，A2，...，Am 是非空跨 t 交集族，则∑1≤i≤m|Ai|≤max{(nk)-∑1≤i≤t-1(ki)(n-kk-i)+m-1,mM(n,k,t)}，其中 M(n,k,t) 是 ([n]k) 的最大 t 交集族的大小。此外，还描述了达到上限的极值族的特征。

引用次数: 0

Avoiding intersections of given size in finite affine spaces AG(n,2) 在有限仿射空间 AG(n,2) 中避免给定大小的交集

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-24 DOI: 10.1016/j.jcta.2024.105959

Benedek Kovács , Zoltán Lóránt Nagy

We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size

m \in [0, 2^{n}]

of the n-dimensional binary affine space

AG (n, 2)

. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in k-dimensional affine subspaces are unavoidable in all m-element point sets in the n-dimensional affine space.

We also show constructions of point sets for which the intersection sizes with k-dimensional affine subspaces take values from a set of a small size compared to

2^{k}

. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.

Keywords: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.

我们研究了 k 维仿射子空间与 n 维二元仿射空间 AG(n,2) 大小为 m∈[0,2n] 的点集的交集大小集。按照厄尔多斯、富雷迪、罗斯柴尔德和 T. 索斯的主题，我们部分确定了 k 维仿射子空间中的哪些局部密度在 n 维仿射空间的所有 m 元素点集中是不可避免的。这些都是由仿射子空间和所谓的子空间规避集建立起来的。同时，我们改进了关于子空间逃避集的已知上界，并应用了关于数字的规范带符号数字（CSD）表示的结果。关键词：不可避免、仿射子空间、逃避集、随机方法、规范带符号数字系统。

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