Journal of Combinatorial Theory Series A最新文献

英文中文

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-08-18 DOI: 10.1016/j.jcta.2025.106101

Yannik Eikmeier, Pamela Fleischmann, Mitja Kulczynski, Dirk Nowotka, Max Wiedenhöft

A prefix normal word is a binary word whose prefixes contain at least as many 1s as any of its factors of the same length. Introduced by Fici and Lipták in 2011, the notion of prefix normality has been, thus far, only defined for words over the binary alphabet. In this work we investigate a generalisation for finite words over arbitrary finite alphabets, namely weighted prefix normality. We prove that weighted prefix normality is more expressive than binary prefix normality. Furthermore, we investigate the existence of a weighted prefix normal form, since weighted prefix normality comes with several new peculiarities that did not already occur in the binary case. We characterise these issues and finally present a standard technique to obtain a generalised prefix normal form for all words over arbitrary, finite alphabets. Additionally, we show a collection of results for the language of those prefix normal forms and extend the connection to Lyndon words and pre-necklaces to the general alphabet.

前缀正常字是一个二进制字，其前缀至少包含与其长度相同的任何因数相同数量的1。2011年，Fici和Lipták引入了前缀正态性的概念，到目前为止，前缀正态性的概念仅用于二进制字母表上的单词。在这项工作中，我们研究了任意有限字母上有限词的泛化，即加权前缀正态性。证明了加权前缀正态性比二元前缀正态性更具表现力。此外，我们研究了加权前缀正规形式的存在性，因为加权前缀正规形式带来了一些新的特性，这些特性在二进制情况下没有出现。我们描述了这些问题，最后提出了一种标准技术来获得任意有限字母上所有单词的广义前缀范式。此外，我们展示了这些前缀正规形式的语言的结果集合，并将与Lyndon单词和pre-项链的连接扩展到一般字母表。

引用次数: 0

Congruences for the smallest parts function associated with ω(q) 与ω(q)相关的最小部分函数的同余

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-09-02 DOI: 10.1016/j.jcta.2025.106106

Renrong Mao

Let

{spt}_{ω} (n)

denote the smallest parts function associated with

ω (q)

. Congruences for

{spt}_{ω} (n)

modulo 5 are first obtained by Andrews, Dixit and Yee. Later, Wang and Yang established two families of congruences for

{spt}_{ω} (n)

modulo powers of 5. More recently, Smoot provided another proof of these congruences and both of the two proofs utilize the Atkin operator

U_{5}

. In this paper, applying the Hecke operators, we obtain congruences for

{spt}_{ω} (n)

modulo powers of primes

ℓ \geq 5

设sptω(n)表示与ω(q)相关的最小部分函数。sptω(n)模5的同余式首先由Andrews， Dixit和Yee得到。后来，Wang和Yang建立了sptω(n) 5的模幂的两个同余族。最近，Smoot提供了这些同余的另一个证明，这两个证明都使用了Atkin算子U5。本文应用Hecke算子，得到了素数≥5的sptω(n)模幂的同余。

引用次数: 0

Level of regions for deformed braid arrangements 变形编织排列区域的水平

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-06-04 DOI: 10.1016/j.jcta.2025.106077

Yanru Chen , Houshan Fu , Suijie Wang , Jinxing Yang

<div><div>This paper primarily investigates a specific type of deformation of the braid arrangement in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, denoted by <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math>. Let <math><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup><mo>)</mo></math> be the number of regions of level l in <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math> and <math><msub><mrow><mi>R</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>;</mo><mi>x</mi><mo>)</mo></math> the corresponding exponential generating function. Using the weighted digraph model introduced by Hetyei, we establish a bijection between regions of level l in <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math> and valid m-acyclic weighted digraphs on the vertex set <math><mo>[</mo><mi>n</mi><mo>]</mo></math> with exactly l strong components. Based on this bijection, we obtain that the sequence <math><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>;</mo><mi>x</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>;</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>⋯</mo></math> is of binomial type. In addition, the values <math><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup><mo>)</mo></math> provide a combinatorial interpretation for the coefficients when the characteristic polynomial of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math> is expanded in terms of <math><mo>(</mo><mtable><mtr><mtd><mi>t</mi></mtd></mtr><mtr><mtd><mi>l</mi></mtd></mtr></mtable><mo>)</mo></math>. In particular, if <math><mi>n</mi><mo>≥</mo><mn>2</mn></math> and <math><mi>A</mi><mo>=</mo><mo>[</mo><mo>−</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo><mo>∩</mo><mi>Z</mi></math> for non-negative integers a and b with <math><mi>b</mi><mo>−</mo><mi>a</mi><mo>≥</mo><mi>n</mi><mo>−</mo><mn>1</mn></math>, we show that the characteristic polynomial of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math> has a single real root 0 of multiplicity one when n is odd, and has one more real root <math><mfrac><mrow><mi>n</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math> of multiplicity one whe

本文主要研究了Rn中编织排列的一种特定变形类型，用AnA表示。设rl（AnA）为AnA中第1层区域的个数，rl（A;x）为对应的指数生成函数。利用Hetyei引入的加权有向图模型，我们在顶点集[n]上建立了AnA中第1层区域与具有1个强分量的有效m-无环加权有向图之间的双射。基于该双射，我们得到序列R1(A;x),R2(A;x)，⋯是二项型。此外，当AnA的特征多项式以（tl）展开时，rl（AnA）值提供了系数的组合解释。特别地，如果n≥2且对于b−A≥n−1的非负整数A和b， A=[−A,b]∩Z，我们证明了当n为奇数时，AnA的特征多项式有一个重数为1的实根0，当n为偶数时，AnA的特征多项式还有一个重数为1的实根n(A +b+1)2。

引用次数: 0

Intersection-union families Intersection-union家庭

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-08-14 DOI: 10.1016/j.jcta.2025.106100

Peter Frankl , Jian Wang

Let

2^{[n]}

denote the power set of the n-set

[n] = {1, 2, \dots, n}

. For positive integers

n, p, q

n \geq p + q

let

m (n, p, q)

denote the maximum of

| F |

for a family

F \subset 2^{[n]}

satisfying

| F \cap G | \geq p

and

| F \cup G | \leq n - q

for all

F, G \in F

. The exact value of

m (n, p, q)

has been known for half a century in the case

p = 1

q = 1

. Bang, Sharp and Winkler determined it in the case

n - p - q \leq 3

. The aim of the present paper is to establish the exact value of

m (n, p, q)

for

n \geq {(n - p - q + 1)}^{3}

and also for

n - p - q = 4

设2[n]表示n-集合[n]={1,2，…，n}的幂集。对于正整数n，p,q, n≥p+q，令m（n,p,q）表示族F∧2[n]满足|F∩G|≥p且对所有F，G∈F满足|F∪G|≤n−q的最大值|F|。在p=1或q=1的情况下，m（n,p,q）的确切值已经知道了半个世纪。Bang， Sharp和Winkler在n−p−q≤3的情况下确定了它。本文的目的是建立当n≥(n−p−q+1)3和n−p−q=4时m（n,p,q）的精确值。

引用次数: 0

Maximum Erdős-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension 偶数维向量空间中最大Erdős-Ko-Rado室集及其反设计

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-08-14 DOI: 10.1016/j.jcta.2025.106098

Philipp Heering , Jesse Lansdown , Klaus Metsch

<div><div>A chamber of the vector space <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math> is a set <math><mo>{</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></math> of subspaces of <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math> where <math><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊂</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊂</mo><mo>…</mo><mo>⊂</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math> and <math><mi>dim</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>i</mi></math> for <math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></math>. By <math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math> we denote the graph whose vertices are the chambers of <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math> with two chambers <math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>{</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></math> and <math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mo>{</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></math> adjacent in <math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math>, if <math><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><mo>=</mo><mo>{</mo><mn>0</mn><mo>}</mo></math> for <math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></math>. The Erdős-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of <math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math>. The independence number of this graph was determined in [5] for n even and given a subspace P of dimension one, the set of all chambers whose subspaces of dimension <math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math> contain P attains the bound. The dual example of course also attains th

向量空间Fqn的一个室是Fqn的子空间{S1，…，Sn−1}的集合，其中S1∧S2∧…∧Sn−1，且对于i=1，…，n−1,dim (Si)=i。通过Γn(q)表示顶点为Fqn的腔室，并且在Γn(q)中相邻两个腔室C1={S1，…，Sn−1}和C2={T1，…，Tn−1}的图，如果Si∩Tn−i={0}，对于i=1，…，n−1。关于腔室的Erdős-Ko-Rado问题相当于确定Γn(q)的独立集的结构。在[5]中对n偶确定图的独立性数，并给定1维的子空间P，其n2维的子空间包含P的所有室的集合达到界。对偶例子当然也得到了边界。它在[5]中保持开放，不管这些是否都是极大独立集。利用[6]对该图最小特征值的特征空间的描述，证明了对于足够大的q， Fqn室的一个Erdős-Ko-Rado定理，给出了对于偶数n的肯定答案。

引用次数: 0

Proof of Lew's conjecture on the spectral gaps of simplicial complexes 卢关于简单配合物谱隙猜想的证明

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-06-18 DOI: 10.1016/j.jcta.2025.106091

Xiongfeng Zhan, Xueyi Huang, Huiqiu Lin

As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let X be a simplicial complex on n vertices, and let

X (k)

denote the set of all k-dimensional simplices of X. The k-th spectral gap

μ_{k} (X)

is the smallest eigenvalue of the reduced k-dimensional Laplacian of X. For any

k \geq - 1

, Lew (2020) [24] established a lower bound for

μ_{k} (X)

μ_{k} (X) \geq (d + 1) (\min_{σ \in X (k)} \deg_{X} (σ) + k + 1) - d n \geq (d + 1) (k + 1) - d n,

where

\deg_{X} (σ)

and d denote the degree of σ in X and the maximal dimension of a missing face of X, respectively. In this paper, we identify the unique simplicial complex that achieves the lower bound of the k-th spectral gap,

(d + 1) (k + 1) - d n

, for some k, thereby confirming a conjecture proposed by Lew.

单纯复形的组合拉普拉斯算子作为图拉普拉斯算子在高维上的推广，越来越受到人们的关注。设X是有n个顶点的简单复形，设X(k)表示X的所有k维简单形的集合。第k个谱间隙μk(X)是X的k维拉普拉斯约简的最小特征值。对于任意k≥- 1，Lew(2020)[24]建立了μk(X)的下界：μk(X)≥(d+1)(minσ∈X(k)∑degX (σ)+k+1)−dn≥(d+1)(k+1)−dn，其中degX （σ）和d分别表示X中σ的阶数和X缺失面的最大维数。在本文中，我们确定了唯一的简单配合物，它达到k的第k谱间隙的下界，（d+1）(k+1)−dn，从而证实了Lew提出的一个猜想。

引用次数: 0

Avoiding short progressions in Euclidean Ramsey theory 在欧几里得拉姆齐理论中避免短级数

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-06-12 DOI: 10.1016/j.jcta.2025.106080

Gabriel Currier , Kenneth Moore , Chi Hoi Yip

We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if

ℓ_{m}

denotes m collinear points with consecutive points of distance one apart, we say that

E^{n} ↛ (ℓ_{r}, ℓ_{s})

if there is a red/blue coloring of n-dimensional Euclidean space that avoids red congruent copies of

ℓ_{r}

and blue congruent copies of

ℓ_{s}

. We show that

E^{n} ↛ (ℓ_{3}, ℓ_{20})

, improving the best-known result

E^{n} ↛ (ℓ_{3}, ℓ_{1177})

by Führer and Tóth, and also establish

E^{n} ↛ (ℓ_{4}, ℓ_{14})

and

E^{n} ↛ (ℓ_{5}, ℓ_{8})

in the spirit of the classical result

E^{n} ↛ (ℓ_{6}, ℓ_{6})

due to Erdős et al. We also show a number of similar 3-coloring results, as well as

E^{n} ↛ (ℓ_{3}, α ℓ_{6889})

, where α is an arbitrary positive real number. This final result answers a question of Führer and Tóth in the positive.

在欧几里得拉姆齐理论中，我们提供了一个构造着色避免短单色算术级数的一般框架。具体地说，如果lm表示m个距离为1的连续点的共线点，我们说En ø （lr, ls）如果n维欧几里德空间的红/蓝着色避免了lr的红色同余拷贝和ls的蓝色同余拷贝。我们通过 hrer和Tóth证明了En倍受（3,1177），改进了最著名的结果En倍受（3,1177），并根据Erdős等人的经典结果En倍受（6,1 6）的精神建立了En倍受（4,1 14）和En倍受（5,1 8）。我们还展示了一些类似的3-着色结果，以及En ø （l3,α l6889），其中α是任意正实数。这个最终结果肯定地回答了一个关于 hrer和Tóth的问题。

引用次数: 0

Real toric manifolds associated with chordal nestohedra 与弦网状体相关的实环流形

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-08-26 DOI: 10.1016/j.jcta.2025.106102

Suyoung Choi, Younghan Yoon

This paper investigates the rational Betti numbers of real toric manifolds associated with chordal nestohedra. We consider the poset topology of a specific poset induced from a chordal building set, and show its EL-shellability. Based on this, we present an explicit description using alternating

B

-permutations for a chordal building set

B

, transforming the computing Betti numbers into a counting problem. This approach allows us to compute the a-number of a finite simple graph through permutation counting when the graph is chordal. In addition, we provide detailed computations for specific cases such as real Hochschild varieties corresponding to Hochschild polytopes.

本文研究了与弦网状体相关的实环流形的有理数Betti数。我们考虑了由弦建筑集导出的特定偏序集的偏序集拓扑，并证明了它的EL-shellability。在此基础上，对弦建筑集B给出了交替B-置换的显式描述，将计算Betti数转化为计数问题。这种方法允许我们通过排列计数来计算有限简单图的a数，当图是弦态时。此外，我们还对Hochschild多面体对应的真实Hochschild变种等具体情况进行了详细的计算。

引用次数: 0

The Frankl-Pach upper bound is not tight for any uniformity 对于任何均匀性，Frankl-Pach上界都是不紧的

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-06-05 DOI: 10.1016/j.jcta.2025.106078

Gennian Ge , Zixiang Xu , Chi Hoi Yip , Shengtong Zhang , Xiaochen Zhao

For any positive integers

n \geq d + 1 \geq 3

, what is the maximum size of a

(d + 1)

-uniform set system in

[n]

with VC-dimension at most d? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound

(\begin{matrix} n \\ d \end{matrix})

via an elegant algebraic proof. Surprisingly, in 2007, Mubayi and Zhao showed that when n is sufficiently large and d is a prime power, the Frankl-Pach upper bound is not tight. They also remarked that their method requires d to be a prime power, and asked for new ideas to improve the Frankl-Pach upper bound without extra assumptions on n and d.

In this paper, we provide an improvement for any

d \geq 2

and

n \geq 2 d + 2

, which demonstrates that the long-standing Frankl-Pach upper bound

(\begin{matrix} n \\ d \end{matrix})

is not tight for any uniformity. Our proof combines a simple yet powerful polynomial method and structural analysis.

对于任意正整数n≥d+1≥3，[n]中vc维不超过d的(d+1)-一致集系统的最大尺寸是多少？1984年，Frankl和Pach开始了对这个基本问题的研究，并通过一个优雅的代数证明给出了上界（nd）。令人惊讶的是，在2007年，Mubayi和Zhao证明了当n足够大且d是素数幂时，Frankl-Pach上界并不紧。他们还指出，他们的方法要求d是素数幂，并要求新的想法来改进Frankl-Pach上界，而不需要对n和d进行额外的假设。在本文中，我们提供了对任意d≥2和n≥2d+2的改进，这证明了长期存在的Frankl-Pach上界（nd）对于任何均匀性都是不严格的。我们的证明结合了一个简单而强大的多项式方法和结构分析。

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引用次数: 0

Perfect codes of bi-Cayley graphs 双凯利图的完美码

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-01 Epub Date: 2025-06-11 DOI: 10.1016/j.jcta.2025.106079

Yan Wang, Kai Yuan, Jian-Xun Li

A bi-Cayley graph is a graph that has a semiregular group H of automorphisms having exactly two orbits on vertices, and it is called an algebraically Cayley graph if its full automorphism group contains a regular subgroup G such that H is a subgroup of G. An independent vertex subset of a graph is called a perfect code if each vertex outside of this subset is adjacent to exactly one vertex in it. In this paper, we give a necessary and sufficient condition for a bi-Cayley graph to be an algebraically Cayley graph, and perfect codes of such bi-Cayley graphs can be determined by the theory of perfect codes in Cayley graphs. Equivalent conditions for subsets to be perfect codes of regular (in terms of graph theory) bi-Cayley graphs are also given.

双凯莱图是一个图，它有一个由恰好两个轨道的自同构组成的半正则群H，如果它的完全自同构群包含一个正则子群G，使得H是G的子群，那么它就被称为代数凯莱图。一个图的独立顶点子集，如果这个子集之外的每个顶点都恰好相邻于其中的一个顶点，就被称为完美码。本文给出了双Cayley图是代数Cayley图的一个充分必要条件，并利用Cayley图的完全码理论确定了双Cayley图的完全码。给出了正则（图论上的）双凯利图的子集为完美码的等价条件。

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