Journal of Combinatorial Theory Series A最新文献

英文中文

Unique representations of integers by linear forms 用线性形式表示整数的唯一形式

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-01 Epub Date: 2025-01-17 DOI: 10.1016/j.jcta.2025.106007

Sándor Z. Kiss , Csaba Sándor

Let

k \geq 2

be an integer and let A be a set of nonnegative integers. For a k-tuple of positive integers

\underline{λ} = (λ_{1}, \dots, λ_{k})

with

1 \leq λ_{1} < λ_{2} < \dots < λ_{k}

, we define the additive representation function

R_{A, \underline{λ}} (n) = | {(a_{1}, \dots, a_{k}) \in A^{k} : λ_{1} a_{1} + \dots + λ_{k} a_{k} = n} |

. For

k = 2

, Moser constructed a set A of nonnegative integers such that

R_{A, \underline{λ}} (n) = 1

holds for every nonnegative integer n. In this paper we characterize all the k-tuples

\underline{λ}

and the sets A of nonnegative integers with

R_{A, \underline{λ}} (n) = 1

for every integer

n \geq 0

设k≥2为整数，设A为一组非负整数。正整数的k-tupleλ_ =(λ1,…,λk) 1≤λ1 & lt;λ2 & lt;⋯& lt;λk,我们定义添加剂表示函数RA,λ_ (n) = | {(a1,…,正义与发展党)∈ak:λ1 a1 +⋯+λ谷湖= n} |。对于k=2， Moser构造了一个非负整数集合a，使得RA，λ_(n)=1对每一个非负整数n都成立。本文刻画了所有k元组λ_和RA，λ_(n)=1对每一个整数n≥0的非负整数集合a。

引用次数: 0

Multivariate P- and/or Q-polynomial association schemes 多元P-和/或q -多项式关联方案

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-01 Epub Date: 2025-02-27 DOI: 10.1016/j.jcta.2025.106025

Eiichi Bannai , Hirotake Kurihara , Da Zhao , Yan Zhu

The classification problem of P- and Q-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of P- and Q-polynomial association schemes to multivariate cases, namely to consider higher rank P- and Q-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definitions nor results. Very recently, Bernard, Crampé, d'Andecy, Vinet, and Zaimi [4], defined bivariate P-polynomial association schemes, as well as bivariate Q-polynomial association schemes. In this paper, we study these concepts and propose a new modified definition concerning a general monomial order, which is more general and more natural and also easy to handle. We prove that there are many interesting families of examples of multivariate P- and/or Q-polynomial association schemes.

P-和q -多项式关联方案的分类问题一直是代数组合学中的核心问题之一。一些作者已经尝试将P-和q -多项式关联方案的概念推广到多元情况，即考虑更高阶的P-和q -多项式关联方案，但迄今为止似乎既没有非常完善的定义也没有结果。最近，Bernard, crampaud, d'Andecy, Vinet, and Zaimi[4]，定义了二元p -多项式关联格式，以及二元q -多项式关联格式。本文对这些概念进行了研究，提出了一个更一般、更自然、更易于处理的关于一般单阶的新的修正定义。我们证明了有许多有趣的多元P-和/或q -多项式关联方案的例子族。

引用次数: 0

Truncated forms of MacMahon's q-series MacMahon q级数的截短形式

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-01 Epub Date: 2025-02-24 DOI: 10.1016/j.jcta.2025.106020

Mircea Merca

<div><div>In 1920, Percy Alexander MacMahon defined the partition generating functions<math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋯</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math> and<math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><mi>k</mi></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋯</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math> which have since played an important rol

1920 年，珀西-亚历山大-麦克马洪定义了分区生成函数 Ak(q)：=∑0<n1<n2<⋯<nkqn1+n2+⋯+nk(1-qn1)2(1-qn2)2⋯(1-qnk)2 和 Ck(q)：=∑0<n1<n2<⋯<nkq2n1+2n2+⋯+2nk-k(1-q2n1-1)2(1-q2n2-1)2⋯(1-q2nk-1)2，它们在组合数学中发挥了重要作用。对于每一个非负整数 k，乔治-安德鲁斯（George E. Andrews）和西蒙-罗斯（Simon C. F. Rose）证明了 Ak(q)可以用分区的生成函数来表示，其中每一部分可以用三种不同颜色中的一种来着色，而 Ck(q)可以用过分区的生成函数来表示。最近，对于每个非负整数 k，Ken Ono 和 Ajit Singh 证明了 Ak(q)、Ak+1(q)、Ak+2(q)......给出了每个部分可以用三种不同颜色中的一种着色的 n 的分区数的生成函数，而 Ck(q)、Ck+1(q)、Ck+2(q)......给出了 n 的过度分区数的生成函数。本文还介绍了一些悬而未决的问题。

引用次数: 0

Simple geometric mitosis 简单几何有丝分裂

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-01 Epub Date: 2025-02-24 DOI: 10.1016/j.jcta.2025.106022

Valentina Kiritchenko

We construct simple geometric operations on faces of the Cayley sum of two polytopes. These operations can be thought of as convex geometric counterparts of divided difference operators in Schubert calculus. We show that these operations give a uniform construction of Knutson–Miller mitosis in the type A and Fujita mitosis in the type C on Kogan faces of Gelfand–Zetlin polytopes.

在两个多面体的Cayley和的面上构造了简单的几何运算。这些操作可以被认为是舒伯特微积分中差除算子的凸几何对应物。我们证明了这些操作在Gelfand-Zetlin多面体的Kogan面上给出了a型的Knutson-Miller有丝分裂和C型的Fujita有丝分裂的统一结构。

引用次数: 0

On common energies and sumsets 在共同的能量和日落

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-01 Epub Date: 2025-02-24 DOI: 10.1016/j.jcta.2025.106026

Shkredov I.D.

We obtain a polynomial criterion for a set to have a small doubling in terms of the common energy of its subsets.

我们得到了一个多项式标准，即一个集合在其子集合的公共能量方面有一个小的加倍。

引用次数: 0

On recursive constructions for 2-designs over finite fields 有限域上2-设计的递归结构

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-01 Epub Date: 2025-01-20 DOI: 10.1016/j.jcta.2025.106006

Xiaoran Wang, Junling Zhou

This paper concentrates on recursive constructions for 2-designs over finite fields. In 1998, Itoh presented a powerful recursive construction: for certain index λ, if there exists a Singer cycle invariant 2-

{(l, 3, λ)}_{q}

design, then there also exists an SL

(m, q^{l})

invariant 2-

{(m l, 3, λ)}_{q}

design for all integers

m \geq 3

. We investigate the

GL (m, q^{l})

-incidence matrix between 2-subspaces and k-subspaces of

GF {(q)}^{m l}

with

m \geq 2

and

k \geq 3

in this work. As a generalization of Itoh's construction, the important case of

m = 2

is supplemented and a doubling construction is established for 2-

{(l, 3, λ)}_{q}

designs over finite fields. As a further generalization, a product construction is developed for q-analogs of group divisible designs (q-GDDs). For general block dimension

k \geq 3

, several new infinite families of q-GDDs are constructed. As applications, plenty of new infinite families of 2-designs over finite fields are constructed.

本文主要研究有限域上2-设计的递归结构。1998年，Itoh提出了一个强大的递归构造：对于某些指标λ，如果存在Singer循环不变量2-(l,3,λ)q设计，那么对于所有整数m≥3，也存在SL（m,ql）不变量2-(ml,3,λ)q设计。本文研究了GF(q)ml中m≥2和k≥3的2-子空间和k-子空间之间的GL(m,ql)-关联矩阵。作为Itoh构造的推广，补充了m=2的重要情况，并建立了有限域上2-(1,3,λ)q设计的双重构造。作为进一步推广，提出了群可分设计q-类似物的产品结构。对于一般块维数k≥3，构造了几个新的q- gdd无限族。作为应用，在有限域上构造了许多新的2-设计无限族。

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引用次数: 0

Structure of Terwilliger algebras of quasi-thin association schemes 拟薄关联格式的Terwilliger代数的结构

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-01 Epub Date: 2025-02-24 DOI: 10.1016/j.jcta.2025.106024

Zhenxian Chen , Changchang Xi

We show that the Terwilliger algebra of a quasi-thin association scheme over a field is always a quasi-hereditary cellular algebra in the sense of Cline-Parshall-Scott and of Graham-Lehrer, respectively, and that the basic algebra of the Terwilliger algebra is the dual extension of a star with all arrows pointing to its center if the field has characteristic 2. Thus many homological and representation-theoretic properties of these Terwilliger algebras can be determined completely. For example, the Nakayama conjecture holds true for Terwilliger algebras of quasi-thin association schemes.

我们证明了场上的拟薄关联格式的Terwilliger代数分别是Cline-Parshall-Scott和Graham-Lehrer意义上的拟遗传元代数，并且如果场具有特征2，则Terwilliger代数的基本代数是一颗所有箭头都指向其中心的星的对偶扩展。从而可以完全确定这些Terwilliger代数的许多同调性质和表示论性质。例如，Nakayama猜想对拟薄关联格式的Terwilliger代数成立。

引用次数: 0

Regular ovoids and Cameron-Liebler sets of generators in polar spaces 极空间中的正则卵圆和Cameron-Liebler生成集

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-07-01 Epub Date: 2025-02-25 DOI: 10.1016/j.jcta.2025.106029

Maarten De Boeck , Jozefien D'haeseleer , Morgan Rodgers

Cameron-Liebler sets of generators in polar spaces were introduced a few years ago as natural generalisations of the Cameron-Liebler sets of subspaces in projective spaces. In this article we present the first two constructions of non-trivial Cameron-Liebler sets of generators in polar spaces. Also regular m-ovoids of k-spaces are introduced as a generalization of m-ovoids of polar spaces. They are used in one of the aforementioned constructions of Cameron-Liebler sets.

极空间中的卡梅隆-利伯勒生成器集是几年前作为投影空间中子空间的卡梅隆-利伯勒集的自然广义而提出的。在这篇文章中，我们首次提出了极空间中非难卡梅隆-利伯勒生成器集的两个构造。此外，还介绍了 k 空间的正则 m-ovoids 作为极空间 m-ovoids 的广义。它们被用于上述卡梅隆-利伯勒集合的一个构造中。

引用次数: 0

Cayley extensions of maniplexes and polytopes 复形和多面体的Cayley扩展

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-05-01 Epub Date: 2024-12-18 DOI: 10.1016/j.jcta.2024.106000

Gabe Cunningham , Elías Mochán , Antonio Montero

A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define

M

to be a Cayley extension of

K

if the facets of

M

are isomorphic to

K

and if some subgroup of the automorphism group of

M

acts regularly on the facets of

M

. We show that many natural extensions in the literature on maniplexes and polytopes are in fact Cayley extensions. We also describe several universal Cayley extensions. Finally, we examine the automorphism group and symmetry type graph of Cayley extensions.

曲面上的自同构群在其顶点上有规则作用的子群的映射称为Cayley映射。这里我们把这个概念推广到复形和多面体。如果M的面与K同构，并且M的自同构群的某子群规律地作用于M的面，我们定义M是K的Cayley扩展。我们证明了文献中许多关于复形和多面体的自然扩展实际上是Cayley扩展。我们还描述了几个通用的Cayley扩展。最后，我们研究了Cayley扩展的自同构群和对称型图。

引用次数: 0

Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions 西顿集，瘦集，和向量布尔函数的非线性

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-05-01 Epub Date: 2024-12-19 DOI: 10.1016/j.jcta.2024.106001

Gábor P. Nagy

The vectorial nonlinearity of a vector-valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager, and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet established a lower bound in terms of differential uniformity. In this paper, we improve Carlet's lower bound. Our approach is based on the fact that the level sets of a vectorial Boolean function are thin sets. In particular, level sets of APN functions are Sidon sets, hence the Liu-Mesnager-Chen conjecture predicts that in

F_{2}^{n}

, there should be Sidon sets of size at least

2^{n / 2} + 1

for all n. This paper provides an overview of the known large Sidon sets in

F_{2}^{n}

, and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane.

向量值函数的向量非线性是它到仿射函数集合的距离。2017年，Liu、Mesnager和Chen推测了向量线性的一般上界。最近，Carlet建立了微分均匀性的下界。本文改进了Carlet下界。我们的方法是基于这样一个事实，即向量布尔函数的水平集是瘦集。特别是，APN函数的水平集是Sidon集，因此，Liu-Mesnager-Chen猜想预测，在F2n中，对于所有n，应该存在大小至少为2n/2+1的Sidon集。本文概述了F2n中已知的大Sidon集，并检验了由有限仿射平面的双曲线和椭圆导出的大Sidon集的完备性。

引用次数: 0

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