European Journal of Combinatorics最新文献

英文中文

A proof of some conjectures of Garvan on partitions rank and crank inequalities 关于分区秩不等式和曲柄不等式的Garvan猜想的证明

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-09 DOI: 10.1016/j.ejc.2025.104253

Renrong Mao, Jie Huang, Fan Yang

In 1988, Garvan made conjectures on inequalities satisfied by ranks and cranks modulo 5 and 7. We obtain improvements to two of these inequalities in this paper.

1988年，Garvan对以5和7为模的秩和曲柄所满足的不等式作了猜想。本文对其中两个不等式作了改进。

引用次数: 0

On two conjectures of Shallit about Thue–Morse-like sequences Shallit关于Thue-Morse-like序列的两个猜想

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-09 DOI: 10.1016/j.ejc.2025.104250

Lubomíra Dvořáková , Savinien Kreczman , Edita Pelantová

We study a class of infinite words

x_{k}

k \in N, k \geq 1

, recently introduced by J. Shallit. This class includes the Thue–Morse sequence

x_{1}

, the Fibonacci–Thue–Morse sequence

x_{2}

, and the Allouche–Johnson sequence

x_{3}

. Shallit stated and for

k = 3

proved two conjectures on properties of

x_{k}

. The first conjecture concerns the factor complexity, the second one the critical exponent of these words. We confirm the validity of both conjectures for every

k

我们研究一类无限词xk， k∈N,k≥1，最近由J. Shallit引入。这个类包括tue - morse序列x1， fibonacci - tue - morse序列x2和Allouche-Johnson序列x3。当k=3时，证明了关于xk性质的两个猜想。第一个猜想与因子复杂性有关，第二个猜想与这些词的临界指数有关。我们对每一个k确认两个猜想的有效性。

引用次数: 0

Growth rates of permutations with given descent or peak set 给定下降集或峰值集的排列生长速率

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-09-30 DOI: 10.1016/j.ejc.2025.104246

Mohamed Omar, Justin M. Troyka

<div><div>Given a set <math><mrow><mi>I</mi><mo>⊆</mo><mi>N</mi></mrow></math>, consider the sequences <math><mrow><mrow><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>}</mo></mrow><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math> where for any <math><mi>n</mi></math>, <math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow></mrow></math> and <math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow></mrow></math> respectively count the number of permutations in the symmetric group <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> whose descent set (respectively peak set) is <math><mrow><mi>I</mi><mo>∩</mo><mrow><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></mrow></mrow></math>. We investigate the growth rates <math><mrow><mo>gr</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msup><mrow><mfenced><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>/</mo><mi>n</mi><mo>!</mo></mrow></mfenced></mrow><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msup></mrow></math> and <math><mrow><mo>gr</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msup><mrow><mfenced><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>/</mo><mi>n</mi><mo>!</mo></mrow></mfenced></mrow><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msup></mrow></math> over all <math><mrow><mi>I</mi><mo>⊆</mo><mi>N</mi></mrow></math>. Our main contributions are two-fold. Firstly, we prove that the numbers <math><mrow><mo>gr</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow></mrow></math> over all <math><mrow><mi>I</mi><mo>⊆</mo><mi>N</mi></mrow></math> are exactly the interval <math><mfenced><mrow><mn>0</mn><mo>,</mo><mn>2</mn><mo>/</mo><mi>π</mi></mrow></mfenced></math>. To do so, we construct an algorithm that explicitly builds <math><mi>I</mi></math> for any desired limit <math><mi>L</mi></math> in the interval. Secondly, we prove that the numbers <math><mrow><mo>gr</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow></mrow></math> for periodic sets <math>

给定集合I∩[N−1]，考虑序列{dn(I)}、{pn(I)}，其中对任意N、dn(I)、pn(I)分别计算对称群Sn中下降集（峰值集）为I∩[N−1]的置换个数。我们研究了增长率grdn(I)=limn→∞dn(I)/n！1/n和grpn(I)=limn→∞pn(I)/n！1/n除以所有I个天大的n。我们的主要贡献有两方面。首先，证明了在所有I个≤N上的数grdn(I)正好是区间0,2/π。为此，我们构造了一个算法，该算法对区间内任何期望的极限L显式地构造I。其次，证明了周期集I≤N的数grpn(I)在0,1/33中形成一个稠密集。我们通过显式地找到，对于区间内任意规定的L，其相应增长率任意接近于L的集合I来做到这一点。

引用次数: 0

On the biases and asymptotics of partitions with finite choices of parts 部分选择有限的分区的偏置与渐近性

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-09-22 DOI: 10.1016/j.ejc.2025.104245

Jiyou Li, Sicheng Zhao

Biases in integer partitions have been studied recently. For three disjoint subsets

R, S, I

of positive integers, let

p_{R S I} (n)

be the number of partitions of

n

with parts from

R \cup S \cup I

and

p_{R > S, I} (n)

be the number of such partitions with a greater number of parts in

R

than that in

S

. In this paper, in the case that

R, S, I

are finite, we obtain an explicit formula of the asymptotic ratio of

p_{R > S, I} (n)

p_{R S I} (n)

. The key technique for computing this ratio is to estimate a partition number at the volume of a certain polytope. A conjecture is proposed in the case that

R, S

are certain infinite arithmetic progressions.

近年来，人们对整数分区中的偏差进行了研究。对于正整数的三个不相交子集R，S，I，设pRSI(n)为n的部分来自R∪S∪I和pR>；S的分区数，I(n)为R中的部分数大于S中的部分数的分区数。本文在R，S，I是有限的情况下，得到了pR>；S，I(n)与pRSI(n)的渐近比的显式公式。计算该比率的关键技术是在某个多面体的体积上估计分区数。在R，S是某些无限等差数列的情况下，提出了一个猜想。

引用次数: 0

Higher q-continued fractions 高q连分数

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-09-20 DOI: 10.1016/j.ejc.2025.104244

Amanda Burcroff , Nicholas Ovenhouse , Ralf Schiffler , Sylvester W. Zhang

We introduce a

q

-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the

q

-rational numbers of Morier-Genoud and Ovsienko. They are defined as ratios of generating functions for

P

-partitions on certain posets. We give matrix formulas for computing them, which generalize previous results in the

q = 1

case. We also show that certain properties enjoyed by the

q

-rationals are also satisfied by our higher versions.

我们引入了由最后三位作者（与Gregg Musiker一起）在之前的工作中引入的高连分数的q-类比，它同时是Morier-Genoud和Ovsienko的q-有序数的推广。它们被定义为特定偏置集上p分区生成函数的比率。我们给出了计算它们的矩阵公式，推广了前人在q=1情况下的结果。我们还证明了q-有理所具有的某些性质也被更高的版本所满足。

引用次数: 0

A counterexample to the Ross–Yong conjecture for Grothendieck polynomials Grothendieck多项式的Ross-Yong猜想的一个反例

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-09-17 DOI: 10.1016/j.ejc.2025.104241

Colleen Robichaux

We give a minimal counterexample for a conjecture of Ross and Yong (2015) which proposes a K-Kohnert rule for Grothendieck polynomials. We conjecture a revised version of this rule. We then prove both rules hold in the 321-avoiding case.

我们给出了Ross和Yong（2015）猜想的最小反例，该猜想提出了格罗滕迪克多项式的K-Kohnert规则。我们推测这条规则的修订版本。然后，我们证明了这两个规则在321避免情况下都成立。

引用次数: 0

Polynomial expressions for the dimensions of the representations of symmetric groups and restricted standard Young tableaux 对称群和限制标准杨氏表的表示维数的多项式表达式

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-09-17 DOI: 10.1016/j.ejc.2025.104242

Avichai Cohen, Shaul Zemel

Given a partition

λ

of a number

k

, it is known that by adding a long line of length

n - k

, the dimension of the associated representation of

S_{n}

is an integer-valued polynomial of degree

k

n

. We show that its expansion in the binomial basis is bounded by the length of

λ

, and that the resulting coefficient of index

h

, with alternating signs, counts the standard Young tableaux of shape

λ

in which a given collection of consecutive

h

numbers lie in increasing rows. We also construct bijections in order to demonstrate explicitly that this number is indeed independent of the set of consecutive

h

numbers used.

给定一个分区数k的λ,众所周知,通过添加一长串长度n−k, Sn的维度关联的表示是一个整数值k次多项式在n。我们证明其在二项式的扩张基础由λ的长度有限,由此产生的系数指数h,交变信号,计算标准的年轻的舞台造型的形状λ给定集合的连续h数量在增加行。为了明确地证明这个数确实独立于所使用的连续h数的集合，我们还构造了双射。

引用次数: 0

Saturation results around the Erdős–Szekeres problem 饱和度是围绕Erdős-Szekeres问题产生的

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-09-16 DOI: 10.1016/j.ejc.2025.104236

Gábor Damásdi , Zichao Dong , Manfred Scheucher , Ji Zeng

In this paper, we consider saturation problems related to the celebrated Erdős–Szekeres convex polygon problem. For each

n \geq 7

, we construct a planar point set of size

(7 / 8) \cdot 2^{n - 2}

which is saturated for convex

n

-gons. That is, the set contains no

n

points in convex position while the addition of any new point creates such a configuration. This demonstrates that the saturation number is smaller than the Ramsey number for the Erdős–Szekeres problem. The proof also shows that the original Erdős–Szekeres construction is indeed saturated. Our construction is based on a similar improvement for the saturation version of the cups-versus-caps theorem. Moreover, we consider the generalization of the cups-versus-caps theorem to monotone paths in ordered hypergraphs. In contrast to the geometric setting, we show that this abstract saturation number is always equal to the corresponding Ramsey number.

本文考虑与著名的Erdős-Szekeres凸多边形问题相关的饱和问题。对于每个n≥7，我们构造一个大小为（7/8）·2n−2的平面点集，该点集对于凸n-gon是饱和的。也就是说，该集合不包含n个处于凸位置的点，而添加任何新点都会创建这样一个构型。这表明饱和数小于Erdős-Szekeres问题的Ramsey数。证明还表明，原来的Erdős-Szekeres结构确实是饱和的。我们的构造是基于杯子对帽子定理的饱和版本的类似改进。此外，我们考虑了cups- vs -caps定理在有序超图单调路径上的推广。与几何设置相反，我们证明了这个抽象饱和数总是等于相应的拉姆齐数。

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引用次数: 0

Odd-Ramsey numbers of complete bipartite graphs 完全二部图的奇拉姆齐数

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-09-13 DOI: 10.1016/j.ejc.2025.104235

Simona Boyadzhiyska , Shagnik Das , Thomas Lesgourgues , Kalina Petrova

In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs

H

K_{n}

, defined as the minimum number of colours needed to colour the edges of

K_{n}

so that every copy of a graph

H \in H

intersects some colour class in an odd number of edges. In this paper, we focus on complete bipartite graphs. First, we completely resolve the problem when

H

is the family of all spanning complete bipartite graphs on

n

vertices. We then focus on its subfamilies, that is,

{K_{t, n - t} : t \in T}

for a fixed set of integers

T \subseteq [⌊ n / 2 ⌋]

. We prove that the odd-Ramsey problem is equivalent to determining the maximum dimension of a linear binary code avoiding codewords of given weights, and leverage known results from coding theory to deduce asymptotically tight bounds in our setting. We conclude with bounds for the odd-Ramsey numbers of fixed (that is, non-spanning) complete bipartite subgraphs.

在他对图码的研究中，Alon引入了Kn中图族H的奇拉姆齐数的概念，定义为为Kn的边上色所需的最小颜色数，使得图H∈H的每一个副本与某个颜色类在奇数条边上相交。本文主要讨论完全二部图。首先，我们完全解决了H是n个顶点上的所有生成完全二部图的族的问题。然后,我们专注于其亚科,{Kt, n−t: t∈t}一组固定的整数t⊆[⌊n / 2⌋]。我们证明了奇拉姆齐问题等价于确定一个避免给定权重码字的线性二进制码的最大维数，并利用编码理论的已知结果在我们的设置中推导出渐近紧界。我们得到了固定（即非生成）完全二部子图的奇拉姆齐数的界。

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引用次数: 0

Homotopy types of Hom complexes of graph homomorphisms whose codomains are square-free 图同态上域为无平方的homo复形的同伦类型

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-09-12 DOI: 10.1016/j.ejc.2025.104238

Soichiro Fujii , Kei Kimura , Yuta Nozaki

Given finite simple graphs

G

and

H

, the Hom complex

Hom (G, H)

is a polyhedral complex having the graph homomorphisms

G \to H

as the vertices. We determine the homotopy type of each connected component of

Hom (G, H)

when

H

is square-free, meaning that it does not contain the 4-cycle graph

C_{4}

as a subgraph. Specifically, for a connected

G

and a square-free

H

, we show that each connected component of

Hom (G, H)

is homotopy equivalent to a wedge sum of circles. We further show that, given any graph homomorphism

f : G \to H

to a square-free

H

, one can determine the homotopy type of the connected component of

Hom (G, H)

containing

f

algorithmically.

给定有限简单图G和H， Hom复形Hom（G，H）是一个顶点为图同态G→H的多面体复形。当H是无平方时，我们确定了hm （G，H）的每个连通分量的同伦类型，这意味着它不包含4循环图C4作为子图。具体地说，对于连通G和无平方H，我们证明了hm （G，H）的每个连通分量同伦等价于圆的楔形和。进一步证明，给定任意图同态f:G→H到一个无平方H，可以用算法确定homm （G，H）包含f的连通分量的同伦类型。

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引用次数: 0

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数据库

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

期刊

European Journal of Combinatorics

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