Journal of Combinatorial Theory Series A最新文献

英文中文

Intersection-union families Intersection-union家庭

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-14 DOI: 10.1016/j.jcta.2025.106100

Peter Frankl , Jian Wang

Let

2^{[n]}

denote the power set of the n-set

[n] = {1, 2, \dots, n}

. For positive integers

n, p, q

n \geq p + q

let

m (n, p, q)

denote the maximum of

| F |

for a family

F \subset 2^{[n]}

satisfying

| F \cap G | \geq p

and

| F \cup G | \leq n - q

for all

F, G \in F

. The exact value of

m (n, p, q)

has been known for half a century in the case

p = 1

q = 1

. Bang, Sharp and Winkler determined it in the case

n - p - q \leq 3

. The aim of the present paper is to establish the exact value of

m (n, p, q)

for

n \geq {(n - p - q + 1)}^{3}

and also for

n - p - q = 4

设2[n]表示n-集合[n]={1,2，…，n}的幂集。对于正整数n，p,q, n≥p+q，令m（n,p,q）表示族F∧2[n]满足|F∩G|≥p且对所有F，G∈F满足|F∪G|≤n−q的最大值|F|。在p=1或q=1的情况下，m（n,p,q）的确切值已经知道了半个世纪。Bang， Sharp和Winkler在n−p−q≤3的情况下确定了它。本文的目的是建立当n≥(n−p−q+1)3和n−p−q=4时m（n,p,q）的精确值。

引用次数: 0

The Weisfeiler-Leman stabilization of a tree 树的Weisfeiler-Leman稳定化

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-13 DOI: 10.1016/j.jcta.2025.106099

Jing Xu , Tatsuro Ito , Shuang-Dong Li

For the Weisfeiler-Leman stabilization, we introduce a concept, which we call the coherent length, to measure how long it takes. We show that the coherent length is at most 8 for trees, using the structures of their T-algebras and of the centralizer algebras of their automorphism groups.

对于Weisfeiler-Leman稳定化，我们引入了一个概念，我们称之为相干长度，来测量它需要多长时间。利用树的t代数及其自同构群的中心化代数的结构，证明了树的相干长度最多为8。

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 existence of m-Haar graphical representations m-Haar图形表示的存在性

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-08-05 DOI: 10.1016/j.jcta.2025.106096

Jia-Li Du , Yan-Quan Feng , Binzhou Xia , Da-Wei Yang

Extending the well-studied concept of graphical regular representations to bipartite graphs, a Haar graphical representation (HGR) of a group G is a bipartite graph whose automorphism group is isomorphic to G and acts semiregularly with the orbits giving the bipartition. The question of which groups admit an HGR was inspired by a closely related question of Estélyi and Pisanski in 2016, as well as Babai's work in 1980 on poset representations, and has been recently solved by Morris and Spiga. In this paper, we introduce the m-Haar graphical representation (m-HGR) as a natural generalization of HGR to m-partite graphs for

m \geq 2

, and explore the existence of m-HGRs for any fixed group. This inquiry represents a more robust version of the existence problem of GmSRs as addressed by Du, Feng and Spiga in 2020. Our main result is a complete classification of finite groups G without m-HGRs.

将已被广泛研究的图形正则表示的概念推广到二部图，群G的Haar图形表示（HGR）是自同构群与G同构并与给出二分的轨道半正则作用的二部图。哪些群体承认HGR的问题受到了est和Pisanski在2016年提出的一个密切相关的问题的启发，以及Babai在1980年对posset表示的研究，莫里斯和斯皮加最近解决了这个问题。本文引入m- haar图表示（m-HGR）作为m≥2时m- haar图表示对m-部图的自然推广，并探讨了m-HGR对任意固定群的存在性。这项研究代表了杜、冯和斯皮加在2020年提出的gmsr存在问题的一个更强大的版本。我们的主要结果是没有m- hgr的有限群G的完全分类。

引用次数: 0

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。

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