Journal of Combinatorial Theory Series A最新文献

英文中文

On nontrivial cross-t-intersecting families 关于非平凡的交叉交集族

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-01 DOI: 10.1016/j.jcta.2025.106095

Dongang He , Anshui Li , Biao Wu , Huajun Zhang

Two families

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

and

B \subseteq (\begin{matrix} [n] \\ ℓ \end{matrix})

are called nontrivial cross-t-intersecting if

| A \cap B | \geq t

for all

A \in A

B \in B

and

| ⋂_{A \in A \cup B} A | < t

. In this paper we will determine the upper bound of

| A | | B |

for nontrivial cross-t-intersecting families

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

and

B \subseteq (\begin{matrix} [n] \\ ℓ \end{matrix})

for positive integers n, k, ℓ and t such that

n \geq \max {(t + 1) (k - t + 1), (t + 1) (ℓ - t + 1)}

and

t \geq 3

. The structures of the extremal families attaining the upper bound are also characterized. As a byproduct of the main result in this paper, one product version of Erdős–Ko–Rado Theorem for two families of cross-t-intersecting can be easily obtained which gives a confirmative answer to one conjecture by Tokushige.

对于所有A∈A， B∈B, |A∈A∪BA|<t，当|A∩B|≥t时，称两个族A （[n]k）和B （[n] r）为非平凡正交相交。在正整数n、k、r、t的条件下，确定非平凡正交族A （[n]k）和B （[n] r）的|A||B|的上界，使n≥max (t+1)(k−t+1),(t+1)(r−t+1)}, t≥3。对达到上界的极族结构也进行了表征。作为本文主要结果的副产品，我们可以很容易地得到两族交叉相交的Erdős-Ko-Rado定理的一个乘积版本，从而证实了Tokushige的一个猜想。

引用次数: 0

General Theta function identities 一般的函数恒等式

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-22 DOI: 10.1016/j.jcta.2025.106094

Sun Kim

Ramanujan's modular equations are closely associated with partition identities. In particular, the modular equations of prime degrees

3, 5, 7, 11

, 23 and the corresponding partition identities are of very elegant forms. These five modular equations were extensively generalized by Warnaar and the present author in the form of general theta function identities. In this paper, we provide further general theta function identities and present many partition identities as special cases.

拉马努金的模方程与分拆恒等式密切相关。特别地，素数阶3、5、7、11、23的模方程和相应的分拆恒等式具有非常优美的形式。这五个模方程被Warnaar和本作者以一般函数恒等式的形式广泛推广。本文进一步给出了一般的函数恒等式，并给出了一些特殊的划分恒等式。

引用次数: 0

Stirling permutation codes. II 斯特林排列码。2

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-11 DOI: 10.1016/j.jcta.2025.106093

Shi-Mei Ma , Hao Qi , Jean Yeh , Yeong-Nan Yeh

In the context of Stirling polynomials, Gessel and Stanley introduced Stirling permutations, which have attracted extensive attention over the past decades. Recently, we introduced Stirling permutation codes and provided numerous equidistribution results as applications. The purpose of the present work is to further analyze Stirling permutation codes. First, we derive an expansion formula expressing the joint distribution of the types A and B descent statistics over the hyperoctahedral group, and we also find an interlacing property involving the zeros of its coefficient polynomials. Next, we prove a strong connection between signed permutations in the hyperoctahedral group and Stirling permutations. We also study unified generalizations of the trivariate second-order Eulerian and ascent-plateau polynomials. Using Stirling permutation codes, we provide expansion formulas for eight-variable and seventeen-variable polynomials, which imply several e-positive expansions and clarify the connection among several statistics. Our results generalize the results of Bóna, Chen-Fu, Dumont, Haglund-Visontai, Janson and Petersen.

在斯特林多项式的背景下，Gessel和Stanley引入了斯特林排列，在过去的几十年里引起了广泛的关注。近年来，我们引入了Stirling排列码，并提供了大量的等分布结果作为应用。本研究的目的是进一步分析斯特林排列码。首先，我们导出了A型和B型下降统计量在高八面体群上的联合分布的展开式，并得到了涉及其系数多项式零点的交错性质。接下来，我们证明了高八面体群中的符号置换与斯特林置换之间的紧密联系。我们还研究了三元二阶欧拉多项式和上升平台多项式的统一推广。利用Stirling排列码，给出了8变量多项式和17变量多项式的展开式，其中蕴涵了若干e正展开式，并阐明了若干统计量之间的联系。我们的结果推广了Bóna、Chen-Fu、Dumont、Haglund-Visontai、Janson和Petersen的结果。

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引用次数: 0

Combinatorics on bi-γ-positivity of 1/k-Eulerian polynomials 1/k-欧拉多项式双γ正性的组合

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-02 DOI: 10.1016/j.jcta.2025.106092

Sherry H.F. Yan , Xubo Yang , Zhicong Lin

The

1 / k

-Eulerian polynomials

A_{n}^{(k)} (x)

were introduced as ascent polynomials over k-inversion sequences by Savage and Viswanathan. The bi-γ-positivity of the

1 / k

-Eulerian polynomials

A_{n}^{(k)} (x)

was known but to give a combinatorial interpretation of the corresponding bi-γ-coefficients still remains open. The study of the theme of bi-γ-positivity from a purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-γ-coefficients of

A_{n}^{(k)} (x)

by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps:

•
construct a bijection between k-Stirling permutations and certain forests that are named increasing pruned even k-ary forests;
•
introduce a generalized Foata–Strehl action on increasing pruned even k-ary trees which implies the longest ascent-plateau polynomials over k-Stirling permutations with initial letter 1 are γ-positive, a result that may have independent interest;
•
develop two crucial transformations on increasing pruned even k-ary forests to conclude our combinatorial interpretation.

1/k欧拉多项式An(k)(x)由Savage和Viswanathan作为k-反转序列上的上升多项式引入。1/k-欧拉多项式An(k)(x)的双γ正性是已知的，但给出相应的双γ系数的组合解释仍然是开放的。从纯组合的角度研究双γ正性的主题是由Athanasiadis提出的。本文利用有序标记森林模型，给出了An(k)(x)的双γ-系数的组合解释。我们的组合方法包括三个主要步骤：•在k-Stirling排列和某些被命名为增加修剪偶数k-ary森林的森林之间构造一个双射；•在增加修剪偶数k-ary树上引入一个广义的fota - strehl作用，该作用表明k-Stirling排列上的首字母为1的最长上升-高原多项式是γ-正的。•发展两个关键的转变，增加修剪甚至k-ary森林，以结束我们的组合解释。

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引用次数: 0

Harmonic higher and extended weight enumerators 谐波高权重和扩展权重枚举数

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-06-27 DOI: 10.1016/j.jcta.2025.106090

Thomas Britz , Himadri Shekhar Chakraborty , Tsuyoshi Miezaki

In this paper, we present the harmonic generalizations of well-known polynomials of codes over finite fields, namely the higher weight enumerators and the extended weight enumerators, and we derive the correspondences between these weight enumerators. Moreover, we present the harmonic generalization of Greene's Theorem for the higher (resp. extended) weight enumerators. As an application of this Greene's-type theorem, we provide the MacWilliams-type identity for harmonic higher weight enumerators of codes over finite fields. Finally, we use this new identity to give a new proof of the Assmus-Mattson Theorem for subcode supports of linear codes over finite fields using harmonic higher weight enumerators.

本文给出了有限域上众所周知的码多项式的调和推广，即高权枚举数和扩展权枚举数，并推导了这些权枚举数之间的对应关系。此外，我们给出了格林定理在高阶方程上的调和推广。扩展)权重枚举数。作为Greene型定理的一个应用，我们给出了有限域上码的调和高权枚举数的macwilliams型恒等式。最后，我们利用这个新恒等式给出了有限域上线性码的子码支持的Assmus-Mattson定理的一个新的证明。

引用次数: 0

Proof of Lew's conjecture on the spectral gaps of simplicial complexes 卢关于简单配合物谱隙猜想的证明

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-06-18 DOI: 10.1016/j.jcta.2025.106091

Xiongfeng Zhan, Xueyi Huang, Huiqiu Lin

As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let X be a simplicial complex on n vertices, and let

X (k)

denote the set of all k-dimensional simplices of X. The k-th spectral gap

μ_{k} (X)

is the smallest eigenvalue of the reduced k-dimensional Laplacian of X. For any

k \geq - 1

, Lew (2020) [24] established a lower bound for

μ_{k} (X)

μ_{k} (X) \geq (d + 1) (\min_{σ \in X (k)} \deg_{X} (σ) + k + 1) - d n \geq (d + 1) (k + 1) - d n,

where

\deg_{X} (σ)

and d denote the degree of σ in X and the maximal dimension of a missing face of X, respectively. In this paper, we identify the unique simplicial complex that achieves the lower bound of the k-th spectral gap,

(d + 1) (k + 1) - d n

, for some k, thereby confirming a conjecture proposed by Lew.

单纯复形的组合拉普拉斯算子作为图拉普拉斯算子在高维上的推广，越来越受到人们的关注。设X是有n个顶点的简单复形，设X(k)表示X的所有k维简单形的集合。第k个谱间隙μk(X)是X的k维拉普拉斯约简的最小特征值。对于任意k≥- 1，Lew(2020)[24]建立了μk(X)的下界：μk(X)≥(d+1)(minσ∈X(k)∑degX (σ)+k+1)−dn≥(d+1)(k+1)−dn，其中degX （σ）和d分别表示X中σ的阶数和X缺失面的最大维数。在本文中，我们确定了唯一的简单配合物，它达到k的第k谱间隙的下界，（d+1）(k+1)−dn，从而证实了Lew提出的一个猜想。

引用次数: 0

Avoiding short progressions in Euclidean Ramsey theory 在欧几里得拉姆齐理论中避免短级数

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-06-12 DOI: 10.1016/j.jcta.2025.106080

Gabriel Currier , Kenneth Moore , Chi Hoi Yip

We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if

ℓ_{m}

denotes m collinear points with consecutive points of distance one apart, we say that

E^{n} ↛ (ℓ_{r}, ℓ_{s})

if there is a red/blue coloring of n-dimensional Euclidean space that avoids red congruent copies of

ℓ_{r}

and blue congruent copies of

ℓ_{s}

. We show that

E^{n} ↛ (ℓ_{3}, ℓ_{20})

, improving the best-known result

E^{n} ↛ (ℓ_{3}, ℓ_{1177})

by Führer and Tóth, and also establish

E^{n} ↛ (ℓ_{4}, ℓ_{14})

and

E^{n} ↛ (ℓ_{5}, ℓ_{8})

in the spirit of the classical result

E^{n} ↛ (ℓ_{6}, ℓ_{6})

due to Erdős et al. We also show a number of similar 3-coloring results, as well as

E^{n} ↛ (ℓ_{3}, α ℓ_{6889})

, where α is an arbitrary positive real number. This final result answers a question of Führer and Tóth in the positive.

在欧几里得拉姆齐理论中，我们提供了一个构造着色避免短单色算术级数的一般框架。具体地说，如果lm表示m个距离为1的连续点的共线点，我们说En ø （lr, ls）如果n维欧几里德空间的红/蓝着色避免了lr的红色同余拷贝和ls的蓝色同余拷贝。我们通过 hrer和Tóth证明了En倍受（3,1177），改进了最著名的结果En倍受（3,1177），并根据Erdős等人的经典结果En倍受（6,1 6）的精神建立了En倍受（4,1 14）和En倍受（5,1 8）。我们还展示了一些类似的3-着色结果，以及En ø （l3,α l6889），其中α是任意正实数。这个最终结果肯定地回答了一个关于 hrer和Tóth的问题。

引用次数: 0

Perfect codes of bi-Cayley graphs 双凯利图的完美码

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-06-11 DOI: 10.1016/j.jcta.2025.106079

Yan Wang, Kai Yuan, Jian-Xun Li

A bi-Cayley graph is a graph that has a semiregular group H of automorphisms having exactly two orbits on vertices, and it is called an algebraically Cayley graph if its full automorphism group contains a regular subgroup G such that H is a subgroup of G. An independent vertex subset of a graph is called a perfect code if each vertex outside of this subset is adjacent to exactly one vertex in it. In this paper, we give a necessary and sufficient condition for a bi-Cayley graph to be an algebraically Cayley graph, and perfect codes of such bi-Cayley graphs can be determined by the theory of perfect codes in Cayley graphs. Equivalent conditions for subsets to be perfect codes of regular (in terms of graph theory) bi-Cayley graphs are also given.

双凯莱图是一个图，它有一个由恰好两个轨道的自同构组成的半正则群H，如果它的完全自同构群包含一个正则子群G，使得H是G的子群，那么它就被称为代数凯莱图。一个图的独立顶点子集，如果这个子集之外的每个顶点都恰好相邻于其中的一个顶点，就被称为完美码。本文给出了双Cayley图是代数Cayley图的一个充分必要条件，并利用Cayley图的完全码理论确定了双Cayley图的完全码。给出了正则（图论上的）双凯利图的子集为完美码的等价条件。

引用次数: 0

The Frankl-Pach upper bound is not tight for any uniformity 对于任何均匀性，Frankl-Pach上界都是不紧的

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-06-05 DOI: 10.1016/j.jcta.2025.106078

Gennian Ge , Zixiang Xu , Chi Hoi Yip , Shengtong Zhang , Xiaochen Zhao

For any positive integers

n \geq d + 1 \geq 3

, what is the maximum size of a

(d + 1)

-uniform set system in

[n]

with VC-dimension at most d? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound

(\begin{matrix} n \\ d \end{matrix})

via an elegant algebraic proof. Surprisingly, in 2007, Mubayi and Zhao showed that when n is sufficiently large and d is a prime power, the Frankl-Pach upper bound is not tight. They also remarked that their method requires d to be a prime power, and asked for new ideas to improve the Frankl-Pach upper bound without extra assumptions on n and d.

In this paper, we provide an improvement for any

d \geq 2

and

n \geq 2 d + 2

, which demonstrates that the long-standing Frankl-Pach upper bound

(\begin{matrix} n \\ d \end{matrix})

is not tight for any uniformity. Our proof combines a simple yet powerful polynomial method and structural analysis.

对于任意正整数n≥d+1≥3，[n]中vc维不超过d的(d+1)-一致集系统的最大尺寸是多少？1984年，Frankl和Pach开始了对这个基本问题的研究，并通过一个优雅的代数证明给出了上界（nd）。令人惊讶的是，在2007年，Mubayi和Zhao证明了当n足够大且d是素数幂时，Frankl-Pach上界并不紧。他们还指出，他们的方法要求d是素数幂，并要求新的想法来改进Frankl-Pach上界，而不需要对n和d进行额外的假设。在本文中，我们提供了对任意d≥2和n≥2d+2的改进，这证明了长期存在的Frankl-Pach上界（nd）对于任何均匀性都是不严格的。我们的证明结合了一个简单而强大的多项式方法和结构分析。

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引用次数: 0

Level of regions for deformed braid arrangements 变形编织排列区域的水平

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-06-04 DOI: 10.1016/j.jcta.2025.106077

Yanru Chen , Houshan Fu , Suijie Wang , Jinxing Yang

<div><div>This paper primarily investigates a specific type of deformation of the braid arrangement in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, denoted by <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math>. Let <math><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup><mo>)</mo></math> be the number of regions of level l in <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math> and <math><msub><mrow><mi>R</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>;</mo><mi>x</mi><mo>)</mo></math> the corresponding exponential generating function. Using the weighted digraph model introduced by Hetyei, we establish a bijection between regions of level l in <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math> and valid m-acyclic weighted digraphs on the vertex set <math><mo>[</mo><mi>n</mi><mo>]</mo></math> with exactly l strong components. Based on this bijection, we obtain that the sequence <math><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>;</mo><mi>x</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>;</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>⋯</mo></math> is of binomial type. In addition, the values <math><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup><mo>)</mo></math> provide a combinatorial interpretation for the coefficients when the characteristic polynomial of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math> is expanded in terms of <math><mo>(</mo><mtable><mtr><mtd><mi>t</mi></mtd></mtr><mtr><mtd><mi>l</mi></mtd></mtr></mtable><mo>)</mo></math>. In particular, if <math><mi>n</mi><mo>≥</mo><mn>2</mn></math> and <math><mi>A</mi><mo>=</mo><mo>[</mo><mo>−</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo><mo>∩</mo><mi>Z</mi></math> for non-negative integers a and b with <math><mi>b</mi><mo>−</mo><mi>a</mi><mo>≥</mo><mi>n</mi><mo>−</mo><mn>1</mn></math>, we show that the characteristic polynomial of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math> has a single real root 0 of multiplicity one when n is odd, and has one more real root <math><mfrac><mrow><mi>n</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math> of multiplicity one whe

本文主要研究了Rn中编织排列的一种特定变形类型，用AnA表示。设rl（AnA）为AnA中第1层区域的个数，rl（A;x）为对应的指数生成函数。利用Hetyei引入的加权有向图模型，我们在顶点集[n]上建立了AnA中第1层区域与具有1个强分量的有效m-无环加权有向图之间的双射。基于该双射，我们得到序列R1(A;x),R2(A;x)，⋯是二项型。此外，当AnA的特征多项式以（tl）展开时，rl（AnA）值提供了系数的组合解释。特别地，如果n≥2且对于b−A≥n−1的非负整数A和b， A=[−A,b]∩Z，我们证明了当n为奇数时，AnA的特征多项式有一个重数为1的实根0，当n为偶数时，AnA的特征多项式还有一个重数为1的实根n(A +b+1)2。

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全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

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