Journal of Combinatorial Theory Series A最新文献

英文中文

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 : 2026-04-01 Epub 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 : 2026-04-01 Epub 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

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 : 2026-04-01 Epub 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

Scarf complexes of graphs and their powers 图的围巾复合体及其幂

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-03-24 DOI: 10.1016/j.jcta.2026.106191

Sara Faridi, Tài Huy Hà, Takayuki Hibi, Susan Morey

引用次数: 0

Focal-free uniform hypergraphs and codes 无焦均匀超图和代码

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-03-24 DOI: 10.1016/j.jcta.2026.106193

Xinqi Huang, Chong Shangguan, Xiande Zhang, Yuhao Zhao

引用次数: 0

On the smallest antichain that generates an ideal of a given size 在最小的反链上产生一个给定尺寸的理想

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-03-20 DOI: 10.1016/j.jcta.2026.106182

L. Sunil Chandran, Rishikesh Gajjala, Kuldeep S. Meel

引用次数: 0

Exponentiable linear orders need not be transitive 可幂线性阶不一定是可传递的

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-03-18 DOI: 10.1016/j.jcta.2026.106192

Mihir Mittal, Amit Kuber

引用次数: 0

The Schur polynomials in all primitive nth roots of unity 所有原始单位n根的舒尔多项式

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-02-01 Epub Date: 2025-09-18 DOI: 10.1016/j.jcta.2025.106107

Masaki Hidaka, Minoru Itoh

We show that the Schur polynomials in all primitive nth roots of unity are 1, 0, or −1, if n has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the coefficients of the cyclotomic polynomial and its multiplicative inverse. The key to the proof is the concept of a unimodular system of vectors. Namely, this result can be reduced to the unimodularity of the tensor product of two maximal circuits (here we call a vector system a maximal circuit, if it can be expressed as

B \cup {- \sum B}

with some basis B).

我们证明了如果n最多有两个不同的奇素数因子，则所有单位的原始n根的舒尔多项式为1,0或- 1。这个结果可以看作是对分环多项式及其乘法逆的系数性质的推广。证明的关键是向量的非模系统的概念。也就是说，这个结果可以简化为两个极大回路的张量积的单模性（这里我们称一个向量系统为极大回路，如果它可以表示为B∪{−∑B}具有某些基B）。

引用次数: 0

Arithmetic properties of MacMahon-type sums of divisors: The odd case macmahon型除数和的算术性质：奇情况

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-02-01 Epub Date: 2025-09-02 DOI: 10.1016/j.jcta.2025.106105

James A. Sellers , Roberto Tauraso

<div><div>A century ago, P. A. MacMahon introduced two families of generating functions,<span><span><span><math><munder><mo>∑</mo><mrow><mn>1</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>t</mi></mrow></msub></mrow></munder><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><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><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mspace></mspace><mtext> and </mtext><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mn>1</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>t</mi></mrow></msub></mtd></mtr><mtr><mtd><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>t</mi></mrow></msub><mtext> odd</mtext></mrow></mtd></mtr></mtable></mrow></munder><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><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><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo></math></span></span></span> which connect sum-of-divisors functions and integer partitions. These have recently drawn renewed attention. In particular, Amdeberhan, Andrews, and Tauraso extended the first family above by defining<span><span><span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>1</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>t</mi></mrow></msub></mrow></munder><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow><mrow><mn>1</mn><mo>+</mo><mi>a</mi><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><m

一个世纪前，P. A. MacMahon引入了两个生成函数族，∑1≤n1<n2<⋯<nt∏k=1tqnk(1−qnk)2和∑1≤n1<；n2<⋯<ntn1,n2，…，nt奇数∏k=1tqnk(1−qnk)2，它们连接了除数和函数和整数分区。最近，这些问题再次引起了人们的关注。特别是，Amdeberhan， Andrews和Tauraso通过定义t（a,q）扩展了上述第一族：=∑1≤n1<n2<⋯<nt∏k=1tqnk1+aqnk+q2nk，对于a=0，±1，±2，并研究了各种性质，包括Ut（a,q）的幂级数表示的系数满足的一些同余。这些算术方面随后被本工作的作者扩展。这里我们的目标是推广第二种生成函数族，其中的和在奇数上运行，然后应用类似的技术来显示相关幂级数系数的新的无限类拉马努金同余族。

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 : 2026-02-01 Epub 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

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