Journal of Combinatorial Theory Series A最新文献

英文中文

Lambert series and double Lambert series 朗伯级数和二重朗伯级数

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-07-01 Epub Date: 2025-12-11 DOI: 10.1016/j.jcta.2025.106154

Tewodros Amdeberhan , George E. Andrews , Cristina Ballantine

We consider relationships between classical Lambert series, multiple Lambert series and classical q-series of the Rogers-Ramanujan type. We conclude with a contemplation on the Andrews-Dixit-Schultz-Yee conjecture.

研究Rogers-Ramanujan型的经典Lambert级数、多重Lambert级数与经典q级数之间的关系。最后，我们对Andrews-Dixit-Schultz-Yee猜想进行了思考。

引用次数: 0

On edge-primitive Cayley graphs 在边基Cayley图上

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-07-01 Epub Date: 2025-12-22 DOI: 10.1016/j.jcta.2025.106155

Zai Ping Lu, Yue Sun

A graph is said to be edge-primitive if its automorphism group acts primitively on the edge set. In this paper, we investigate finite edge-primitive Cayley graphs of valency no less than 2. An explicit classification of such graphs is obtained in the case where the graphs admit an almost simple edge-primitive automorphism group which contains a regular subgroup on the vertices. This implies that the only edge-primitive Cayley graphs of valency at least 2 defined over simple groups are cycles with prime length and complete graphs. In addition, we also classify those edge-primitive Cayley graphs which are either 2-arc-transitive or of square-free order.

如果图的自同构群基本作用于边集，则称图为边基图。本文研究了价不小于2的有限边基Cayley图。当图承认一个几乎简单的边原自同构群，且该群在顶点上包含正则子群时，得到了这类图的显式分类。这表明在单群上定义的价至少为2的边基Cayley图只有素数长度的环和完全图。此外，我们还对2-弧传递和无平方阶的边基Cayley图进行了分类。

引用次数: 0

Van Lint–MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II 有限域上Cayley图的Van Lint-MacWilliams猜想和最大团，2

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-07-01 Epub Date: 2026-01-19 DOI: 10.1016/j.jcta.2026.106162

Chi Hoi Yip

The well-known Van Lint–MacWilliams' conjecture states that if q is an odd prime power, and

A \subseteq F_{q^{2}}

such that

0, 1 \in A

| A | = q

, and

a - b

is a square for each

a, b \in A

, then A must be the subfield

F_{q}

. This conjecture was first proved by Blokhuis and is often phrased in terms of the maximum cliques in Paley graphs of square order. Previously, Asgarli and the author extended Blokhuis' theorem to a larger family of Cayley graphs. In this paper, we give a new simple proof of Blokhuis' theorem and its extensions. More generally, we show that if

S \subseteq F_{q^{2}}^{⁎}

has small multiplicative doubling, and

A \subseteq F_{q^{2}}

with

0, 1 \in A

| A | = q

, such that

A - A \subseteq S \cup {0}

, then

A = F_{q}

. This new result refines and extends several previous works; moreover, our new approach avoids using heavy machinery from number theory.

著名的Van Lint-MacWilliams猜想认为，如果q是奇质数幂，且A∈Fq2使得0,1∈A, |A|=q，且A - b是每个A， b∈A的平方，则A一定是子域Fq。这个猜想最早是由Blokhuis证明的，并且通常用平方阶Paley图中的最大团来表述。先前，Asgarli和作者将Blokhuis定理推广到更大的Cayley图族。本文给出了Blokhuis定理的一个新的简单证明及其扩展。更一般地说，我们证明了如果S≥≥1，且A≥≥1∈A， |≤A≤|=q，则A≤Fq。这个新结果完善和扩展了以前的几项工作；此外，我们的新方法避免了使用数论中的重型机器。

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引用次数: 0

Complete mappings stabilizing a subgroup and their parities 稳定子群及其对偶的完全映射

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-07-01 Epub Date: 2025-12-05 DOI: 10.1016/j.jcta.2025.106143

Shikang Yu , Tao Feng , Hengrui Liu

A complete mapping of a finite group G is a permutation

ϕ : G \to G

such that

x \mapsto x ϕ (x)

is also a permutation of G. The complete mapping ϕ is called odd (resp. even) when ϕ is an odd (resp. even) permutation. To address whether the permutation groups generated by all complete mappings of a finite group G are “large” and primitive, Bors and Wang (2023) initiated the investigation into which groups admit both even and odd complete mappings. It is conjectured that such groups exist if and only if their Sylow 2-subgroups are trivial or noncyclic, except for some groups of small order. This paper reduces the problem of determining whether an arbitrary finite group admits both even and odd complete mappings to examining (1) the existence of complete mappings that stabilize a Sylow 2-subgroup for every finite simple group of Lie type and every sporadic simple group in the set

{Ly, {Co}_{2}, B, Th, {Fi}_{22}, {Fi}_{23}, {Fi}_{24}^{'}, Ru, O^{'} N, {Co}_{1}, {Co}_{3}, HN, M}

, and (2) the existence of complete mappings that stabilize a subgroup of order

8 p_{0}^{t_{0}}

(for some odd prime

p_{0}

and

t_{0} ⩾ 1

) for every Ree group

{}^{2}G_{2} (3^{2 n + 1})

with

n ⩾ 1

有限群G的完全映射是一个φ:G→G的置换，使得x∑xφ (x)也是G的置换。偶数)，当φ是奇数时。甚至)排列。为了解决由有限群G的所有完备映射生成的置换群是否“大”且原始的问题，Bors和Wang（2023）开始研究哪些群同时存在奇偶完备映射。我们推测，除了一些小阶群外，这样的群当且仅当它们的Sylow 2-子群是平凡的或非循环的。本文将确定任意有限群是否存在奇偶完全映射的问题简化为检验(1)对于集合{Ly，Co2，B,Th,Fi22,Fi23,Fi24 ',Ru,O ' n,Co1,Co3，HN，M}中的每一个Lie型有限简单群和每一个偶然性简单群，是否存在稳定Sylow 2-子群的完全映射；以及(2)对于具有n大于或等于1的每个Ree组G22（32n+1）稳定8p0阶子群（对于一些奇数素数p0和t0小于或等于1）的完整映射的存在。

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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 : 2026-05-01 Epub 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

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 : 2026-05-01 Epub 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

Mean and variance of the longest alternating subsequence in a random separable permutation 随机可分排列中最长交替子序列的均值和方差

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2025-12-31 DOI: 10.1016/j.jcta.2025.106157

Ross G. Pinsky

<div><div>A permutation is separable if it can be obtained from the singleton permutation by iterating direct sums and skew sums. Equivalently, it is separable if and only it avoids the patterns 2413 and 3142. Under the uniform probability on separable permutations of <math><mo>[</mo><mi>n</mi><mo>]</mo></math>, let the random variable <math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math> denote the length of the longest alternating subsequence. Also, let <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo><mo>,</mo><mo>−</mo></mrow></msubsup></math> denote the length of the longest alternating subsequence that begins with an ascent and ends with a descent, and define <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mo>,</mo><mo>+</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo><mo>,</mo><mo>+</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mo>,</mo><mo>−</mo></mrow></msubsup></math> similarly. By symmetry, the first two and the last two of these latter four random variables are equi-distributed. We prove that the expected value of any of these five random variables behaves asymptotically as <math><mo>(</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mi>n</mi><mo>≈</mo><mn>0.5858</mn><mspace></mspace><mi>n</mi></math>. We also obtain the more refined estimates that the expected value of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo><mo>,</mo><mo>−</mo></mrow></msubsup></math> and of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mo>,</mo><mo>+</mo></mrow></msubsup></math> is equal to <math><mo>(</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mi>n</mi><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> and that the expected value of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo><mo>,</mo><mo>+</mo></mrow></msubsup></math> and of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mo>,</mo><mo>−</mo></mrow></msubsup></math> is equal to <math><mo>(</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mi>n</mi><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math>. Finally, we show that the variance of any of the four random variables <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>±</mo

一个排列是可分离的，如果它能从单态排列中通过迭代直接和和和得到。同样，当且仅当它避免了模式2413和3142时，它是可分离的。在可分离排列[n]的均匀概率下，设随机变量An表示最长交替子序列的长度。同样，设An+，−表示以上升开始，以下降结束的最长交替子序列的长度，并类似地定义An−，+,An+,+,An−,−。根据对称性，后四个随机变量的前两个和后两个是等分布的。我们证明了这五个随机变量的期望值的渐近性为（2−2）n≈0.5858n。我们还得到了An+，−和An−，+的期望值等于（2−2）n−14(3−22)+o(1)和An+，+和An−，−的期望值等于（2−2）n+34(3−22)+o(1)的更精确估计。最后，我们证明了任意四个随机变量的方差An±，±的渐近表现为16−1122n≈0.2218n。

引用次数: 0

Corrigendum to “Quantum nilpotent subalgebras of classical quantum groups and affine crystals” [J. Comb. Theory, Ser. A 168 (2019) 219–254] “经典量子群和仿射晶体的量子幂零子代数”的更正[J]。合成杆。理论,爵士。[A] [168 (2019) 219-254]

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2026-01-27 DOI: 10.1016/j.jcta.2026.106161

Il-Seung Jang , Jae-Hoon Kwon

In this corrigendum, we revise [1, Lemma 5.12] and give a revised proof of [1, Lemma 5.14(2)].

在这个勘误表中，我们修正了[1，引理5.12]，并给出了[1，引理5.14(2)]的一个修正证明。

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 : 2026-05-01 Epub 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

Helly numbers for quantitative Helly-type results Helly数用于定量的Helly型结果

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2026-01-16 DOI: 10.1016/j.jcta.2026.106160

Grigory Ivanov , Márton Naszódi

We obtain three Helly-type results. First, we establish a Quantitative Colorful Helly-type theorem with the optimal Helly number 2d concerning the diameter of the intersection of a family of convex bodies. Second, we prove a Quantitative Helly-type theorem with the optimal Helly number

2 d + 1

for the pointwise minimum of logarithmically concave functions. Finally, we present a colorful version of the latter result with Helly number (number of color classes)

3 d + 1

; however, we have no reason to believe that this bound is sharp.

我们得到了三个helly型结果。首先，我们建立了关于一类凸体交点直径的最优Helly数2d的彩色Helly型定量定理。其次，我们证明了对数凹函数的点最小值的最优Helly数2d+1的定量Helly型定理。最后，我们给出了后一种结果的彩色版本，Helly数（颜色类数）3d+1；然而，我们没有理由相信这个界限是尖锐的。

引用次数: 0

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