Journal of Combinatorial Theory Series A最新文献_第3页

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

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

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全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

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Journal of Combinatorial Theory Series A

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