Journal of Combinatorial Theory Series A最新文献_第8页

Flag transitive geometries with trialities and no dualities coming from Suzuki groups 标志传递几何的三角性和没有对偶性来自铃木群

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 10.1016/j.jcta.2025.106033

Dimitri Leemans , Klara Stokes , Philippe Tranchida

Recently, Leemans and Stokes constructed an infinite family of incidence geometries admitting trialities but no dualities from the groups

P S L (2, q)

(where

q = p^{3 n}

with p a prime and

n > 0

a positive integer). Unfortunately, these geometries are not flag transitive. In this paper, we work with the Suzuki groups

S z (q)

, where

q = 2^{2 e + 1}

with e a positive integer and

2 e + 1

is divisible by 3. For any odd integer m dividing

q - 1

,

q + \sqrt{2 q} + 1

or

q - \sqrt{2 q} + 1

(i.e.: m is the order of some non-involutive element of

S z (q)

), we construct geometries of type

(m, m, m)

that admit trialities but no dualities. We then prove that they are flag transitive when

m = 5

, no matter the value of q. These geometries form the first infinite family of incidence geometries of rank 3 that are flag transitive and have trialities but no dualities. They are constructed using chamber systems and the trialities come from field automorphisms. These same geometries can also be considered as regular hypermaps with automorphism group

S z (q)

.

最近，Leemans和Stokes从群PSL(2,q)（其中q=p3n， p为素数，n>；0为正整数）中构造了一个无限族的关联几何，它们只承认试验而不承认对偶。不幸的是，这些几何图形不是标志传递的。本文研究铃木群Sz(q)，其中q=22e+1，且e为正整数，且2e+1能被3整除。对于任意奇数m除q−1，q+2q+1或q−2q+1（即m是Sz(q)的某个非对合元素的阶），我们构造了（m,m,m）型几何，它承认三次性但不承认对偶性。然后证明当m=5时，无论q的值是多少，它们都是标志可传递的。这些几何构成了第一个无限的3级关联几何族，它们是标志可传递的，有三角但没有对偶。它们是用腔室系统构造的，特性来自于场自同构。这些相同的几何也可以被认为是具有自同构群Sz(q)的正则超映射。

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引用次数: 0

Separable elements and splittings in Weyl groups of type B B型Weyl群的可分离元素与分裂

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 10.1016/j.jcta.2025.106021

Ming Liu, Houyi Yu

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair

(X, Y)

of subsets of the symmetric group

S_{n}

, the multiplication map

X \times Y \to S_{n}

is a splitting (i.e., a length-additive bijection) of

S_{n}

if and only if X is the generalized quotient of Y and Y is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type B.

Weyl群中的可分离元素是对称群中著名的可分离置换的推广。Gaetz和Gao证明了对于对称群Sn的任意子集对（X，Y），当且仅当X是Y的广义商，Y是由可分离元素生成的右弱阶主低阶理想时，乘法映射X×Y→Sn是Sn的分裂（即长度加性双射）。他们推测这个结果可以推广到所有有限Weyl群。本文用禁止模式对所有可分和最小不可分有符号置换进行了分类，并证实了B型Weyl群的Gaetz和Gao猜想。

引用次数: 0

A bijection related to Bressoud's conjecture 与布雷苏德的猜想有关的一个问题

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 10.1016/j.jcta.2025.106032

Y.H. Chen, Thomas Y. He

Bressoud introduced the partition function

B (α_{1}, \dots, α_{λ}; η, k, r; n)

, which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition function

B (α_{1}, \dots, α_{λ}; η, k, r; n)

in multi-summation form. In this article, we introduce a bijection related to Bressoud's conjecture. As an application, we give the proof of a companion to the Göllnitz-Gordon identities.

Bressoud引入了配分函数B(α1，…，αλ;η,k,r;n)，用于统计具有一定差异条件的配分数。Bressoud以多重求和形式对配分函数B（α1，…，αλ;η,k,r;n）的生成函数提出了一个猜想。在这篇文章中，我们引入了一个与Bressoud猜想有关的双射。作为一个应用，我们给出了Göllnitz-Gordon身份的一个伴侣的证明。

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 : 2025-02-27 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

On de Bruijn rings and families of almost perfect maps 在德布鲁因环和几乎完美的地图上

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 10.1016/j.jcta.2025.106030

Peer Stelldinger

De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape

(m, n)

appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd

m = n \in {3, 5, 7}

and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These maps capture a large number of patterns, with each appearing at most once. While previous methods for generating such sub-perfect maps cover only a fraction of the possible patterns, we present a construction method for generating almost perfect maps for arbitrary pattern shapes and arbitrary non-prime alphabet sizes, including the above mentioned square tori with odd

m = n \in {3, 5, 7}

as long that the alphabet size is non-prime. This is achieved through the introduction of de Bruijn rings, a minimal-height sub-perfect map and a formalization of the concept of families of almost perfect maps. The generated sub-perfect maps are easily decodable which makes them perfectly suitable for positional coding applications.

De Bruijn tori，或完美映射，是有限字母表中字母的二维周期性数组，其中每种可能的形状模式（m,n）在单个周期内恰好出现一次。虽然某些de Bruijn环面（例如奇数m=n∈{3,5,7}和偶数字母大小的平方环面）的存在性仍未得到解决，但在位置编码等应用中，次完美映射通常是足够的。这些地图捕获了大量的模式，每种模式最多出现一次。虽然以前生成这种次完美映射的方法只覆盖了可能模式的一小部分，但我们提出了一种生成任意模式形状和任意非素数字母表大小的几乎完美映射的构造方法，包括上面提到的奇数m=n∈{3,5,7}的方形环面，只要字母表大小是非素数。这是通过引入de Bruijn环，一个最小高度的次完美地图和几乎完美地图族概念的形式化来实现的。生成的次完美地图很容易解码，这使得它们非常适合位置编码应用。

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引用次数: 0

An imperceptible connection between the Clebsch–Gordan coefficients of Uq(sl2) and the Terwilliger algebras of Grassmann graphs Uq（sl2）的Clebsch-Gordan系数与Grassmann图的Terwilliger代数之间难以察觉的联系

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 10.1016/j.jcta.2025.106028

Hau-Wen Huang

<div><div>The Clebsch–Gordan coefficients of <math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra <math><mi>H</mi></math> into <math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>. Let Ω denote a finite set of size D and <math><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></math> denote the power set of Ω. It is generally known that <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math> supports a <math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>-module. Let k denote an integer with <math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi></math> and fix a k-element subset <math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math> of Ω. By identifying <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math> with <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi><mo>∖</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msup><mo>⊗</mo><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msup></math> this induces a <math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>-module structure on <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math> denoted by <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math>. Pulling back via ♮ the <math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>-module <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math> forms an <math><mi>H</mi></math>-module. When <math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi><mo>−</mo><mn>1

U（sl2）的Clebsch-Gordan系数可以用Hahn多项式表示。这一现象可以用从通用哈恩代数H到U(sl2)⊗U（sl2）的代数同态解释。设Ω表示大小为D的有限集，2Ω表示Ω的幂集。众所周知，C2Ω支持U(sl2)-模块。设k为0≤k≤D的整数，固定Ω的一个k元素子集x0。通过将C2Ω与C2Ω∈x0⊗C2x0识别，在C2Ω上得到一个U(sl2)⊗U（sl2）模块结构，表示为C2Ω（x0）。通过缩回调，U(sl2)⊗U(sl2)-模C2Ω（x0）形成h模。当1≤k≤D−1时，h模C2Ω（x0）包涵Johnson图J（D,k）关于x0的Terwilliger代数。这个结果将两个看似无关的主题联系起来：U（sl2）的Clebsch-Gordan系数和Johnson图的Terwilliger代数。不幸的是，在q模拟的情况下，有些步骤失效了。本文通过绕弯路，成功地揭示了Uq（sl2）的Clebsch-Gordan系数与Grassmann图的Terwilliger代数之间不易察觉的联系。

引用次数: 0

More on r-cross t-intersecting families for vector spaces 更多关于向量空间的r- x - t相交族

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-26 DOI: 10.1016/j.jcta.2025.106031

Tian Yao , Dehai Liu , Kaishun Wang

Let V be a finite dimensional vector space over a finite field. Suppose that

F_{1}

,

F_{2}

, …,

F_{r}

are r-cross t-intersecting families of k-subspaces of V. In this paper, we determine the extremal structure when

\prod_{i = 1}^{r} | F_{i} |

is maximum under the condition that

\dim (⋂_{F \in F_{i}} F) < t

for each i.

设V是有限维向量空间在有限域上。设F1， F2，…，Fr是v的k个子空间的r-交叉t-相交族。本文在每个i取dim (F∈FiF)<；t的条件下，确定了∏i=1r|Fi|最大时的极值结构。

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 : 2025-02-25 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

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 : 2025-02-24 DOI: 10.1016/j.jcta.2025.106020

Mircea Merca

In 1920, Percy Alexander MacMahon defined the partition generating functions

A_{k} (q) : = \sum_{0 < n_{1} < n_{2} < \dots < n_{k}} \frac{q^{n_{1} + n_{2} + \dots + n_{k}}}{{(1 - q^{n_{1}})}^{2} {(1 - q^{n_{2}})}^{2} \dots {(1 - q^{n_{k}})}^{2}}

and

C_{k} (q) : = \sum_{0 < n_{1} < n_{2} < \dots < n_{k}} \frac{q^{2 n_{1} + 2 n_{2} + \dots + 2 n_{k} - k}}{{(1 - q^{2 n_{1} - 1})}^{2} {(1 - q^{2 n_{2} - 1})}^{2} \dots {(1 - q^{2 n_{k} - 1})}^{2}}

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 : 2025-02-24 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。

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Journal of Combinatorial Theory Series A

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