Journal of Combinatorial Theory Series A最新文献_第10页

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

On the number of subsequence sums related to the support of a sequence in finite abelian groups 有限阿贝尔群中与序列支持度相关的子序列和的数目

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-24 DOI: 10.1016/j.jcta.2025.106124

Rui Wang, Han Chao, Jiangtao Peng

Let G be a finite abelian group and S a sequence with elements of G. Let

| S |

denote the length of S and

supp (S)

the set of all the distinct terms in S. For an integer k with

k \in [1, | S |]

, let

Σ_{k} (S) \subset G

denote the set of group elements which can be expressed as a sum of a subsequence of S with length k. Let

Σ (S) = \cup_{k = 1}^{| S |} Σ_{k} (S)

and

Σ_{\geq k} (S) = \cup_{t = k}^{| S |} Σ_{t} (S)

. It is known that if

0 \notin Σ (S)

, then

| Σ (S) | \geq | S | + | supp (S) | - 1

. In this paper, we determine the structure of a sequence S satisfying

0 \notin Σ (S)

and

| Σ (S) | = | S | + | supp (S) | - 1

. As a consequence, we can give a counterexample of a conjecture of Gao, Grynkiewicz, and Xia. Moreover, we prove that if

| S | > k

and

0 \notin Σ_{\geq k} (S) \cup supp (S)

, then

| Σ_{\geq k} (S) | \geq | S | - k + | supp (S) |

. Then we can give an alternative proof of a conjecture of Hamidoune, which was first proved by Gao, Grynkiewicz, and Xia.

设G是一个有限阿贝尔群，S是一个具有G元素的序列，设|S|表示S的长度，supp(S)表示S中所有不同项的集合。对于k∈[1，|S|]的整数k，设Σk(S)∧G表示可以表示为长度为k的S的子序列的和的群元素集合，设Σ(S)=∪k=1|S|Σk(S)和Σ≥k(S)=∪t=k|S|Σt(S)。众所周知,如果0∉Σ(S),然后|Σ(S) |≥| | + |增刊(S) |−1。本文确定了满足0∈Σ(S)和|Σ(S)|=|S|+|supp(S)|−1的序列S的结构。因此，我们可以给出Gao、Grynkiewicz和Xia猜想的一个反例。此外,我们证明如果| |在k和0∉Σ≥k (S)∪增刊(S),然后|Σ≥k (S) |≥|年代|−k + |增刊(S) |。然后，我们可以给出由Gao、Grynkiewicz和Xia首先证明的Hamidoune猜想的另一种证明。

引用次数: 0

Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials 由雅可比函数和特殊对称多项式引起的拟模形式

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-16 DOI: 10.1016/j.jcta.2025.106123

Tewodros Amdeberhan , Leonid G. Fel , Ken Ono

Ramanujan derived a sequence of even weight 2n quasimodular forms

U_{2 n} (q)

from derivatives of Jacobi's weight 3/2 theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series

F (X)

. Using the weight 1 form

θ {(q)}^{2}

and

F (X) = \exp (X / 2)

, we obtain a sequence

{Y_{n} (q)}

of weight n quasimodular forms on

Γ_{0} (4)

whose symmetric function avatars

{\tilde{Y}}_{n} (x^{k})

are the symmetric polynomials

T_{n} (x^{k})

that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the

T_{n} (x^{k})

. Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch

\hat{A}

-genus for spin manifolds, where one identifies power sum symmetric functions

p_{i}

with Pontryagin classes.

Ramanujan从Jacobi的权值3/2函数的导数中导出了一个偶数权值2n的拟模形式U2n(q)的序列。利用该序列的生成函数，可以构造具有最小输入的所有非负整数权的准模形式序列：权1模形式和幂级数F(X)。利用权值为1的形式θ(q)2和F(X)=exp (X/2)，在Γ0(4)上得到了一个权值为n的拟模形式序列{Yn(q)}，其对称函数元Y ~ n（xk）是研究数值半群协同时自然产生的对称多项式Tn（xk）。有了这些信息，我们确定了关于Tn（xk）的两个猜想。最后，我们注意到这些多项式是系统地用自旋流形的Borel-Hirzebruch A -格给出的，其中人们用Pontryagin类识别幂和对称函数pi。

引用次数: 0

Dissection of the quintuple product, with applications 五元积的解剖，及其应用

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-13 DOI: 10.1016/j.jcta.2025.106122

Tim Huber , James Mc Laughlin , Dongxi Ye

This work considers the m-dissection (for

m ≢ 0 (\mod 3)

) of the general quintuple product

Q (z, q) = {(z, q / z, q; q)}_{\infty} {(q z^{2}, q / z^{2}; q^{2})}_{\infty} .

Multiple novel applications arise from this m-dissection. For example, we derive the general partition identity

D_{S} (m n + (m^{2} - 1) / 24) = {(- 1)}^{(m + 1) / 6} b_{m} (n), for all n \geq 0,

where

m \equiv 5 (\mod 6)

is a square-free positive integer relatively prime to 6;

D_{S} (n)

is defined, for S the set of positive integers containing no multiples of m, to be the number of partitions of n into an even number of distinct parts from S minus the number of partitions of n into an odd number of distinct parts from S; and

b_{m} (n)

denotes the number of m-regular partitions of n. The dissections allow us to prove a conjecture of Hirschhorn concerning the

2^{n}

-dissection of

{(q; q)}_{\infty}

, as well as determine the pattern of the sign changes of the coefficients

a_{n}

of the infinite product

(q^{2^{}}

本文考虑一般五元积tq (z,q)=(z,q/z,q;q)∞(qz2,q/z2;q2)∞的m-剖分（对于m > 0(mod3)）。这种m型解剖产生了多种新的应用。例如，我们推导出一般划分恒等式ds (mn+(m2−1)/24)=(−1)(m+1)/6bm(n)，对于所有n≥0，其中m≡5（mod6）是一个相对素数为6的无平方正整数；定义DS(n)，对于S是不含m的正整数集合，等于n被划分为从S出发的偶数个可分离部分的个数减去n被划分为从S出发的奇数个可分离部分的个数；bm(n)表示n的m个正则分割的个数。这些分割证明了Hirschhorn关于(q;q)∞的2n-分割的一个猜想，并确定了无穷积(q2k−1;q2k−1)∞(qp;qp)∞2=∑n=0∞和qn，k≥1，p≥5a素数的系数an的符号变化规律。这涵盖了Bringmann等人最近的结果，对应于k=1和p=5的情况。

引用次数: 0

Near triple arrays 近三元数组

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-02 DOI: 10.1016/j.jcta.2025.106121

Alexey Gordeev, Klas Markström, Lars-Daniel Öhman

We introduce near triple arrays as binary row-column designs with at most two consecutive values for the replication numbers of symbols, for the intersection sizes of pairs of rows, pairs of columns and pairs of a row and a column. Near triple arrays form a common generalization of such well-studied classes of designs as triple arrays, (near) Youden rectangles and Latin squares.

We enumerate near triple arrays for a range of small parameter sets and show that they exist in the vast majority of the cases considered. As a byproduct, we obtain the first complete enumerations of

6 \times 10

triple arrays on 15 symbols,

7 \times 8

triple arrays on 14 symbols and

5 \times 16

triple arrays on 20 symbols.

Next, we give several constructions for families of near triple arrays, and e.g. show that near triple arrays with 3 rows and at least 6 columns exist for any number of symbols. Finally, we investigate a duality between row and column intersection sizes of a row-column design, and covering numbers for pairs of symbols by rows and columns. These duality results are used to obtain necessary conditions for the existence of near triple arrays. This duality also provides a new unified approach to earlier results on triple arrays and balanced grids.

我们引入近三重数组作为二进制行-列设计，对于符号的复制数，对于行对，列对和行与列对的相交大小，具有最多两个连续值。近三组数组形成了诸如三组数组、（近）约登矩形和拉丁正方形等设计的共同概括。我们列举了一系列小参数集的近三重数组，并表明它们存在于所考虑的绝大多数情况下。作为副产品，我们获得了第一个完整的枚举：6×10 15个符号上的三元数组，7×8 14个符号上的三元数组和5×16 20个符号上的三元数组。其次，我们给出了近三列数组族的几种构造，并举例说明了任意数目的符号都存在3行至少6列的近三列数组。最后，我们研究了行-列设计的行和列相交大小之间的对偶性，以及行和列对符号的覆盖数。利用这些对偶结果，得到了近三元数组存在的必要条件。这种对偶性还提供了一种新的统一方法来处理三重阵列和平衡网格的早期结果。

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引用次数: 0

k-Adjoint of hyperplane arrangements 超平面排列的k伴随

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-10-01 DOI: 10.1016/j.jcta.2025.106120

Weikang Liang , Suijie Wang , Chengdong Zhao

In this paper, we introduce the k-adjoint of a given hyperplane arrangement

A

associated with rank-k elements in the intersection lattice

L (A)

, which generalizes the classical adjoint proposed by Bixby and Coullard. The k-adjoint of

A

induces a decomposition of the Grassmannian, which we call the

A

-adjoint decomposition. Inspired by the work of Gelfand, Goresky, MacPherson, and Serganova, we generalize the matroid decomposition and refined Schubert decomposition of the Grassmannian from the perspective of

A

. Furthermore, we prove that these three decompositions are exactly the same decomposition. A notable application involves providing a combinatorial classification of all the k-dimensional restrictions of

A

. Consequently, we establish the antitonicity of some combinatorial invariants, such as Whitney numbers of the first kind and the independence numbers.

本文引入了交格L(a)中与秩k元素相关的超平面排列a的k伴随，推广了Bixby和Coullard提出的经典伴随。A的k伴随引起了格拉斯曼分解，我们称之为A伴随分解。受Gelfand， Goresky, MacPherson， and Serganova的工作启发，我们从a的角度推广了Grassmannian的类矩阵分解和细化了Schubert分解，并证明了这三种分解是完全相同的分解。一个值得注意的应用涉及提供A的所有k维限制的组合分类，因此，我们建立了一些组合不变量的反抗性，如第一类惠特尼数和独立数。

引用次数: 0

Improved asymptotics for moments of reciprocal sums for partitions into distinct parts 分割成不同部分的互易和矩的改进渐近性

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-09-26 DOI: 10.1016/j.jcta.2025.106119

Kathrin Bringmann , Byungchan Kim , Eunmi Kim

In this paper we strongly improve asymptotics for

s_{1} (n)

(respectively

s_{2} (n)

) which sums reciprocals (respectively squares of reciprocals) of parts throughout all the partitions of n into distinct parts. The methods required are much more involved than in the case of usual partitions since the generating functions are not modular and also do not possess product expansions.

本文强烈改进了s1(n)（分别为s2(n)）的渐近性，该渐近性对n的所有划分为不同部分的部分的倒数（分别为倒数的平方）求和。由于生成函数不是模块化的，也不具有乘积展开，因此所需的方法比通常分区的情况要复杂得多。

引用次数: 0

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