Journal of Combinatorial Theory Series A最新文献

英文中文

Superport networks 超级网络

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-12-01 DOI: 10.1016/j.jcta.2025.106134

P. Pylyavskyy , S. Shirokovskikh , M. Skopenkov

We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair. If one sets the voltage difference for each pair, then the incoming currents are uniquely determined. We generalize Kirchhoff's matrix-tree theorem to this setup. Different forests now contribute with different signs, making the proof subtle. In particular, we use the formula for the response matrix minors by R. Kenyon–D. Wilson, determinantal identities, and combinatorial bijections. We introduce superport networks, generalizing both ordinary networks and multiport ones.

我们研究电气工程中常见的多端口网络。它们的边界条件不同于电网络：边界顶点被分成对，每对输入电流的总和被设置为零。如果为每一对设置电压差，那么输入电流是唯一确定的。我们将基尔霍夫矩阵树定理推广到这种情况。不同的森林现在有不同的迹象，使证据变得微妙。特别地，我们使用了R. Kenyon-D的响应矩阵子式。Wilson，行列式恒等式和组合对偶。我们介绍了超级网络，对普通网络和多端口网络进行了推广。

引用次数: 0

Towards odd-sunflowers: Temperate families and lightnings 走向奇异的向日葵：温带科和闪电

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-28 DOI: 10.1016/j.jcta.2025.106133

Jan Petr , Pavel Turek

Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and Pálvölgyi, we initiate the study of temperate families: a family

F \subseteq P ([n])

is said to be temperate if each

A \in F

contains at most

| A |

elements of

F

as a proper subset.

We show that the maximum size of a temperate family is attained by the middle two layers of the hypercube

{0, 1}^{n}

. As a more general result, we obtain that the middle

t + 1

layers of the hypercube maximise the size of a family

F

such that each

A \in F

contains at most

\sum_{j = 1}^{t} (\begin{matrix} | A | \\ j \end{matrix})

elements of

F

as a proper subset. Moreover, we classify all such families consisting of the maximum number of sets.

In the case of intersecting temperate families, we find the maximum size and classify all intersecting temperate families consisting of the maximum number of sets for odd n. We also conjecture the maximum size for even n.

受Frankl、Pach和Pálvölgyi最近引入的奇向日葵的启发，我们发起了对温带家族的研究：如果每个a∈F最多包含F的|00个a |个元素作为适当子集，则称一个家族F∈P（[n]）是温带的。我们证明了温带族的最大尺寸是在超立方体{0,1}n的中间两层。作为一个更一般的结果，我们得到了超立方体的中间t+1层使族F的大小最大化，使得每个a∈F最多包含F的∑j=1t（| a |j）个元素作为一个固有子集。此外，我们对所有由最大数量集合组成的族进行了分类。对于相交的温带族，我们找到了最大的大小，并对奇数n下由最大集合数组成的所有相交的温带族进行了分类。我们还推测了偶数n下的最大大小。

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引用次数: 0

Eulerian-type polynomials over Stirling permutations and box sorting algorithm 斯特林排列上的欧拉型多项式和盒排序算法

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-28 DOI: 10.1016/j.jcta.2025.106132

Shi-Mei Ma , Jun-Ying Liu , Jean Yeh , Yeong-Nan Yeh

It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by many researchers. This paper is divided into three parts. The first part gives a convolution formula for the second-order Eulerian polynomials, which simplifies a result of Gessel. As an application, a determinantal expression for the second-order Eulerian polynomial is obtained. We then investigate a convolution formula of the trivariate second-order Eulerian polynomials. Among other things, by introducing three new statistics: proper ascent-plateau, improper ascent-plateau and trace, we discover that a six-variable enumerative polynomial over restricted Stirling permutations equals a six-variable Eulerian-type polynomial over signed permutations. By special parametrizations, we make use of Stirling permutations to give a unified interpretation of the

(p, q)

-Eulerian polynomials and derangement polynomials of types A and B. The third part presents a box sorting algorithm which leads to a bijection between the terms in the expansion of

{(c D)}^{n} c

and ordered weak set partitions, where c is a smooth function in the indeterminate x and D is the derivative with respect to x. Using a map from ordered weak set partitions to standard Young tableaux, we find an expansion of

{(c D)}^{n} c

in terms of standard Young tableaux. Combining this with context-free grammars, we provide three new interpretations of the second-order Eulerian polynomials.

众所周知，在经典斯特林排列集合上，上升、下降和高原是均匀分布的。其常见的枚举多项式是二阶欧拉多项式，已被许多研究者广泛研究。本文共分为三个部分。第一部分给出了二阶欧拉多项式的卷积公式，简化了Gessel的结果。作为应用，得到了二阶欧拉多项式的行列式。然后我们研究了三元二阶欧拉多项式的卷积公式。此外，通过引入三个新的统计量：适当的上升-高原、不适当的上升-高原和迹，我们发现限制斯特林置换上的六变量枚举多项式等于符号置换上的六变量欧拉型多项式。通过特殊的参数化,我们使用斯特灵的排列给一个统一的解释(p, q)欧拉多项式多项式和错乱的类型a和b,第三部分提出了一个盒子排序算法导致条款之间的双射的扩张(cD)数控,命令弱设置分区,c是一个光滑函数的不定x和D是导数x。使用标准命令弱设置分区地图年轻的场景,我们发现了（cD）nc在标准杨氏表的展开式。结合上下文无关语法，我们提供了三种二阶欧拉多项式的新解释。

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引用次数: 0

Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation 非交换色拟对称函数，Macdonald多项式，和Yang-Baxter方程

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-21 DOI: 10.1016/j.jcta.2025.106130

Jean-Christophe Novelli, Jean-Yves Thibon

As shown in our paper (Novelli and Thibon, 2021 [20]), the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to WQSym, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behavior with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-t version of these noncommutative analogues. For rectangular partitions, their commutative images at

q = 0

appear to coincide with the multi-t Hall-Littlewood functions introduced in (Lascoux et al., 1995 [14]). This leads us to conjecture that for rectangular partitions, multi-t Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.

如我们的论文（Novelli and Thibon, 2021[20]）所示，Shareshian-Wachs的色拟对称函数可以提升到WQSym，即非交换变量中拟对称函数的代数。我们在这里研究了它在经典字母变换中的行为，并提出了一个与Haglund-Wilson公式的非交换版本兼容的Macdonald多项式的非交换模拟。我们还介绍了这些非交换类似物的多t版本。对于矩形分区，它们在q=0处的交换像似乎与（Lascoux et al., 1995[14]）中引入的多t Hall-Littlewood函数一致。这使我们推测，对于矩形分区，多重t Macdonald多项式可以作为Hecke代数的某些Yang-Baxter元素的等变迹得到。我们还推测所有（普通）麦克唐纳多项式都可以用这种方法得到。最后，我们对拟对称色函数的各个方面与Hecke代数中的计算作了一些评述。特别地，我们证明了所有模关系都是由Kazhdan-Lusztig基的乘积公式给出的。

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引用次数: 0

On cover-free families of finite vector spaces 有限向量空间的无复族

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-21 DOI: 10.1016/j.jcta.2025.106131

Yunjing Shan, Junling Zhou

There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptography and communications. This work studies the generalization of cover-free families from sets to finite vector spaces. Let V be an n-dimensional vector space over the finite field

F_{q}

and let

{[\frac{V}{k}]}_{q}

denote the family of all k-dimensional subspaces of V. A family

F \subseteq {[\frac{V}{k}]}_{q}

is called cover-free if there are no three distinct subspaces

F_{0}, F_{1}, F_{2} \in F

such that

F_{0} \leq (F_{0} \cap F_{1}) + (F_{0} \cap F_{2})

. A family

H \subseteq {[\frac{V}{k}]}_{q}

is called a q-Steiner system

S_{q} (t, k, n)

if for every

T \in {[\frac{V}{t}]}_{q}

, there is exactly one

H \in H

such that

T \leq H

. In this paper we investigate cover-free families in the vector space V. Firstly, we determine the maximum size of a cover-free family in

{[\frac{V}{k}]}_{q}

. Secondly, we characterize the structures of all maximum cover-free families which are closely related to q-Steiner systems.

由于有限集的无复族在组合群测试、密码学和通信等领域的广泛应用，有大量的文献研究有限集的无复族。本文研究了无复族从集合到有限向量空间的推广。设V是有限域Fq上的一个n维向量空间，设[Vk]q表示V的所有k维子空间的族。如果不存在三个不同的子空间F0、F1、F2∈F使F0≤(F0∩F1)+(F0∩F2)，则称族F≠Vk。如果对每一个t∈[Vt]q，有一个H∈H使t≤H，则称为q-施泰纳体系Sq（t,k,n）。本文研究了向量空间v中的无覆盖族，首先确定了[Vk]q中无覆盖族的最大大小。其次，我们刻画了与q-Steiner系统密切相关的所有最大无覆盖族的结构。

引用次数: 0

On large Sidon sets 在大型西顿集上

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-20 DOI: 10.1016/j.jcta.2025.106129

Ingo Czerwinski, Alexander Pott

A (binary) Sidon set M is a subset of

F_{2}^{t}

such that the sum of four distinct elements of M is never 0. The goal is to find Sidon sets of large size. In this note we show that the graphs of almost perfect nonlinear (APN) functions with high linearity can be used to construct large Sidon sets. Thanks to recently constructed APN functions

F_{2}^{8} \to F_{2}^{8}

with high linearity, we can construct Sidon sets of size 192 in

F_{2}^{15}

, where the largest sets so far had size 152. Using the inverse and the Dobbertin function also gives larger Sidon sets as previously known. Each of the new large Sidon sets M in

F_{2}^{t}

yields a binary linear code with t check bits, minimum distance 5, and a length not known so far. Moreover, we improve the upper bound for the linearity of arbitrary APN functions.

一个（二进制）西顿集合M是F2t的一个子集，使得M中四个不同元素的和永远不为0。目标是找到大的西顿集。在本文中，我们证明了具有高线性度的几乎完全非线性（APN）函数的图可以用来构造大的西顿集。由于最近构造了具有高线性度的APN函数F28→F28，我们可以在F215中构造大小为192的Sidon集合，其中迄今为止最大的集合大小为152。使用逆函数和Dobbertin函数也可以得到更大的西顿集。F2t中的每个新的大西顿集M产生一个二进制线性码，有t个校验位，最小距离5，长度到目前为止还不知道。此外，我们改进了任意APN函数线性度的上界。

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引用次数: 0

On r-cross t-intersecting families for vector spaces with large product of sizes 具有大尺寸积的向量空间的r-交叉t-相交族

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-19 DOI: 10.1016/j.jcta.2025.106127

Jie Wen, Benjian Lv

Let V be an n-dimensional vector space over a finite field, and

[\begin{matrix} V \\ k \end{matrix}]

denote the set of k-dimensional subspaces of V. We say that

F_{1} \subseteq [\begin{matrix} V \\ k_{1} \end{matrix}], F_{2} \subseteq [\begin{matrix} V \\ k_{2} \end{matrix}], \dots, F_{r} \subseteq [\begin{matrix} V \\ k_{r} \end{matrix}]

are r-cross t-intersecting if

\dim (F_{1} \cap F_{2} \cap \dots \cap F_{r}) \geq t

for all

F_{i} \in F_{i}, i = 1, 2, \dots, r

. The families are trivial if every subspace in those families contains a common specified subspace of dimension t, and are non-trivial otherwise. In this paper, we determine the structure of non-trivial r-cross t-intersecting families with maximum product of their sizes for

r \geq 3

, and give a stability result for

r \geq 4

. To prove these results, we first provide a new lower bound for n, which does not depend on t, ensuring that families maximizing the product of sizes are trivial.

设V为有限域上的n维向量空间，[Vk]表示V的k维子空间的集合。设对于所有Fi∈Fi，i=1,2，…，r，如果dim (F1∩F2∩⋯∩Fr)≥t，则F1≥Vk1，F2≥Vk2，…，Fr≤Vkr，则F1≥Vk1， f≤Vk1, f≤Vk1, f≤Vk2, f≤Vkr, f≤Vkr。如果这些族中的每一个子空间都包含一个维数为t的公共指定子空间，则这些族是平凡的，否则是非平凡的。本文确定了r≥3时具有最大积的非平凡r-交叉t-相交族的结构，并给出了r≥4时的稳定性结果。为了证明这些结果，我们首先为n提供了一个新的下界，它不依赖于t，以确保最大化大小乘积的族是平凡的。

引用次数: 0

Sequences of odd length in strict partitions I: The combinatorics of double sum Rogers-Ramanujan type identities 严格分区中的奇长序列I：双和Rogers-Ramanujan型恒等式的组合

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-18 DOI: 10.1016/j.jcta.2025.106128

Shishuo Fu, Haijun Li

Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum q-series. Equipped with such a combinatorial set-up, we investigate a handful of double sum identities appeared in recent works of Cao-Wang, Wang-Wang, Wei-Yu-Ruan, Andrews-Uncu, Chern, and Wang, finding partition theoretical interpretations to all of these identities, and in most cases supplying Franklin-type involutive proofs. This approach dates back more than a century to P. A. MacMahon's interpretations of the celebrated Rogers-Ramanujan identities, and has been further developed by Kurşungöz in the last decade.

严格划分是根据权重、部件数量和奇数长度序列的数量来列举的。我们把这个三元生成函数写成一个双和q级数。在这样的组合设置下，我们研究了曹旺、王旺、阮维宇、Andrews-Uncu、chen和Wang最近的作品中出现的一些双和恒等式，找到了对所有这些恒等式的分拆理论解释，并在大多数情况下提供了富兰克林式的对合证明。这种方法可以追溯到一个多世纪前P. a . MacMahon对著名的罗杰斯-拉马努金身份的解释，并在过去十年中被Kurşungöz进一步发展。

引用次数: 0

Induced arithmetic removal for partition-regular patterns of complexity 1 复杂度为1的分区规则模式的诱导算法去除

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-04 DOI: 10.1016/j.jcta.2025.106126

V. Gladkova

In 2019, Fox, Tidor and Zhao [7] proved an induced arithmetic removal lemma for linear patterns of complexity 1 in vector spaces over a fixed finite field. With no further assumptions on the pattern, this induced removal lemma cannot guarantee a fully pattern-free recolouring of the space, as some ‘non-generic’ instances must necessarily remain. On the other hand, Bhattacharyya, Fischer, H. Hatami, P. Hatami, and Lovett [3] showed in 2012 that in the case of translation-invariant patterns, it is possible to obtain recolourings that eliminate the given pattern completely, with no exceptions left behind. This paper demonstrates that such complete removal can be achieved for all partition-regular patterns of complexity 1.

在2019年，Fox， Tidor和Zhao[7]证明了在固定有限域上向量空间中复杂度为1的线性模式的诱导算法去除引理。由于没有对图案的进一步假设，这种诱导去除引理不能保证空间完全无图案的重新着色，因为一些“非一般”实例必须保留。另一方面，Bhattacharyya, Fischer， H. Hatami， P. Hatami和Lovett[3]在2012年表明，对于平移不变模式，可以获得完全消除给定模式的再着色，没有任何例外。本文证明了对于复杂度为1的所有分区规则模式都可以实现这种完全去除。

引用次数: 0

Semicomplete multipartite weakly distance-regular digraphs 半完全多部弱距离正则有向图

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-28 DOI: 10.1016/j.jcta.2025.106125

Shuang Li , Yuefeng Yang , Kaishun Wang

A digraph is semicomplete multipartite if its underlying graph is a complete multipartite graph. As a special case of semicomplete multipartite digraphs, Jørgensen et al. [7] initiated the study of doubly regular team tournaments. As a natural extension, we introduce doubly regular team semicomplete multipartite digraphs and show that such digraphs fall into three types. Furthermore, we give a characterization of all semicomplete multipartite commutative weakly distance-regular digraphs.

如果一个有向图的底图是一个完全多部图，那么它就是半完全多部图。作为半完全多部有向图的特例，Jørgensen et al. b[7]发起了双常规团队比赛的研究。作为一种自然推广，我们引入了双正则团队半完全多部有向图，并证明了这类有向图可分为三种类型。进一步，我们给出了所有半完全多部可交换弱距离正则有向图的一个刻画。

引用次数: 0

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