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Chainlink Polytopes and Ehrhart Equivalence 链锁多面体和艾哈特等价性
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2024-02-06 DOI: 10.1007/s00026-023-00683-x
Ezgi Kantarcı Oǧuz, Cem Yalım Özel, Mohan Ravichandran

We introduce a class of polytopes that we call chainlink polytopes and show that they allow us to construct infinite families of pairs of non-isomorphic rational polytopes with the same Ehrhart quasipolynomial. Our construction is based on circular fence posets, a recently introduced class of posets, which admit a non-obvious and nontrivial symmetry in their rank sequences. We show that this symmetry can be lifted to the level of polyhedral models (which we call chainlink polytopes) for these posets. Along the way, we introduce the related class of chainlink posets and show that they exhibit analogous nontrivial symmetry properties. We further prove an outstanding conjecture on the unimodality of rank polynomials of circular fence posets.

我们引入了一类多边形,称之为链环多边形,并证明它们允许我们构建具有相同艾尔哈特准多项式的无限对非同构有理多边形族。我们的构造基于圆形栅栏正方体,这是最近引入的一类正方体,在它们的秩序列中存在一个非显而易见的非难对称性。我们证明,这种对称性可以提升到这些正集的多面体模型(我们称之为链环多面体)的水平。同时,我们还介绍了相关的链环集合类,并证明它们具有类似的非对称对称性。我们还进一步证明了一个关于圆栅栏正方体秩多项式单模态性的杰出猜想。
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引用次数: 0
The Maximum Number of Cliques in Graphs with Bounded Odd Circumference 有界奇数圆周图中的最大聚类数
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2024-01-24 DOI: 10.1007/s00026-023-00682-y
Zequn Lv, Ervin Győri, Zhen He, Nika Salia, Chuanqi Xiao, Xiutao Zhu

In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo’s recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

在这项工作中,我们给出了有界奇数周长图中小群数的尖锐上界。这个广义的图兰型结果是著名的厄尔多斯和加莱定理的扩展,也是罗氏最新结果的加强。对于有界偶数周长的图,同样的约束是李和宁定理的一个微不足道的应用。
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引用次数: 0
A Unified Combinatorial Treatment for Three Classical Truncated Theta Series 三种经典截断θ数列的统一组合处理方法
IF 0.5 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2024-01-23 DOI: 10.1007/s00026-023-00684-w
Andrew Y. Z. Wang, Ang Xiao

There has been a tremendous amount of research on the truncated theta series in the past decade. How can we understand them combinatorially? In this paper, we investigate the truncated theorems of three classical theta series of Euler and Gauss, and provide a unified combinatorial treatment. Meanwhile, we propose a possible and more direct approach to deal with these truncated theorems.

在过去的十年中,对截断的 Theta 数列进行了大量的研究。如何从组合的角度理解它们呢?在本文中,我们研究了欧拉和高斯的三个经典θ级数的截断定理,并提供了统一的组合处理方法。同时,我们提出了一种可能的、更直接的方法来处理这些截断定理。
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引用次数: 0
A Succinct Proof of Defant and Kravitz’s Theorem on the Length of Hitomezashi Loops 德凡特和克拉维茨关于人字形环路长度定理的简明证明
IF 0.5 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2024-01-02 DOI: 10.1007/s00026-023-00681-z
Qiuyu Ren, Shengtong Zhang

We provide a much shorter proof of Defant and Kravitz’s theorem that the length of Hitomezashi loops is congruent to 4 modulo 8. Our novel idea is to consider the length module 8 for Hitomezashi paths that take an excursion in a half-plane region.

我们为迪凡特和克拉维茨的定理提供了一个更简短的证明,即人字桥环路的长度与 4 模 8 全等。我们新颖的想法是考虑在半平面区域内游走的人头桥路径的长度模数 8。
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引用次数: 0
Two Enriched Poset Polytopes 两个丰富的 Poset 多面体
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2023-12-22 DOI: 10.1007/s00026-023-00679-7
Soichi Okada, Akiyoshi Tsuchiya

Stanley introduced and studied two lattice polytopes, the order polytope and chain polytope, associated with a finite poset. Recently, Ohsugi and Tsuchiya introduce an enriched version of them, called the enriched order polytope and enriched chain polytope. In this paper, we give a piecewise-linear bijection between these enriched poset polytopes, which is an enriched analogue of Stanley’s transfer map and bijectively proves that they have the same Ehrhart polynomials. Also, we construct explicitly unimodular triangulations of two enriched poset polytopes, which are the order complexes of graded posets.

斯坦利提出并研究了与有限正集相关的两个格状多面体--阶多面体和链多面体。最近,Ohsugi 和 Tsuchiya 引入了它们的丰富版本,称为丰富阶多胞形和丰富链多胞形。在本文中,我们给出了这些富集正多胞形之间的片线性偏射,这是斯坦利转移映射的富集类似物,并偏射地证明了它们具有相同的艾哈特多项式。此外,我们还构造了两个丰富正多胞形的显式单模三角剖分,它们都是分级正多胞形的阶复数。
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引用次数: 0
Approximate Sampling of Graphs with Near-P-Stable Degree Intervals 近似采样具有近 P 稳定度区间的图
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2023-12-21 DOI: 10.1007/s00026-023-00678-8
Péter L. Erdős, Tamás Róbert Mezei, István Miklós

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller–Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently, Amanatidis and Kleer published a beautiful paper (DOI:10.4230/LIPIcs.STACS.2023.7), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper, we substantially extend their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centered at P-stable degree sequences.

摘要 在一些社会科学、计算机科学、工程学等项目中,对具有给定度序列的图形现实进行近似均匀采样是一项日常任务。一种方法是使用马尔可夫链。关于研究得很透彻的开关马尔可夫链,目前最好的结果是它能在 P 个稳定的度序列上快速混合(见 DOI:10.1016/j.ejc.2021.103421)。切换马尔可夫链不会改变任何度序列。然而,在有些情况下,指定的是度数区间而不是单一的度数序列。(出现这个问题的一个自然场景是对只有部分观测数据的社交网络进行假设检验)。Rechner, Strowick 和 Müller-Hannemann 在 2018 年提出了程度区间马尔科夫链的概念,它使用了三种(分别研究得很好)局部操作(切换、铰链-翻转和切换),并采用了程度序列实现,其中任何两个被审查的序列都具有非常小的坐标距离。最近,Amanatidis 和 Kleer 发表了一篇漂亮的论文(DOI:10.4230/LIPIcs.STACS.2023.7),表明如果序列来自一个非常细的区间系统,而这些区间的中心离一个规则的度数序列不远,那么度数区间马尔可夫链就会迅速混合。在本文中,我们大幅扩展了他们的结果,证明了如果区间以 P 个稳定的度序列为中心,度区间马尔可夫链是快速混合的。
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引用次数: 0
An Asymptotic Lower Bound on the Number of Polyominoes 多面体数量的渐近下限
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2023-12-21 DOI: 10.1007/s00026-023-00675-x
Vuong Bui

Let P(n) be the number of polyominoes of n cells and (lambda ) be Klarner’s constant, that is, (lambda =lim _{nrightarrow infty } root n of {P(n)}). We show that there exist some positive numbers AT, so that for every n

$$begin{aligned} P(n) ge An^{-Tlog n} lambda ^n. end{aligned}$$

This is somewhat a step toward the well-known conjecture that there exist positive (C,theta ), so that (P(n)sim Cn^{-theta }lambda ^n) for every n. In fact, if we assume another popular conjecture that (P(n)/P(n-1)) is increasing, we can get rid of (log n) to have

$$begin{aligned} P(n)ge An^{-T}lambda ^n. end{aligned}$$

Beside the above theoretical result, we also conjecture that the ratio of the number of some class of polyominoes, namely inconstructible polyominoes, over P(n) is decreasing, by observing this behavior for the available values. The conjecture opens a nice approach to bounding (lambda ) from above, since if it is the case, we can conclude that

$$begin{aligned} lambda < 4.1141, end{aligned}$$

which is quite close to the current best lower bound (lambda > 4.0025) and greatly improves the current best upper bound (lambda < 4.5252). The approach is merely analytically manipulating the known or likely properties of the function P(n), instead of giving new insights of the structure of polyominoes. The techniques can be applied to other lattice animals and self-avoiding polygons of a given area with almost no change.

让 P(n) 是 n 个单元的多面体数,而 (lambda ) 是克拉纳常数,即 (lambda =lim _{nrightarrow infty })。根 n ({P(n)})。我们证明存在一些正数 A、T,所以对于每一个 n$$(开始{对齐}P(n) ge An^{-Tlog n}lambda ^n.end{aligned}$$ 其中 ( (lambda)是克拉纳常数,即 ( (lambda =lim _{nrightarrow infty })。root n ({P(n)}的根)。事实上,如果我们假设另一个流行的猜想是(P(n)/P(n-1))是递增的,我们就可以摆脱(log n) 得到$$begin{aligned}。P(n)ge An^{-T}lambda ^n.end{aligned}$$除了上述理论结果,我们还猜想,通过观察可用值的这一行为,某类多角体(即不可构造多角体)的数量与 P(n) 的比率是递减的。这个猜想开辟了一个很好的方法来从上面限定 (lambda ),因为如果是这样的话,我们就可以得出结论:$$begin{aligned}。4.1141, end{aligned}$$这非常接近于当前最好的下界(4.0025),并且大大提高了当前最好的上界(4.5252)。这种方法只是对函数 P(n) 的已知或可能性质进行了分析处理,而不是对多面体的结构提出新的见解。这些技术可以应用于其他格子动物和给定面积的自避让多边形,几乎没有任何变化。
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引用次数: 0
Labeled Chip-Firing on Binary Trees with (2^n-1) Chips 有$2^n-1$$芯片的二叉树上的标签芯片发射
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2023-12-15 DOI: 10.1007/s00026-023-00680-0
Gregg Musiker, Son Nguyen

We study labeled chip-firing on binary trees starting with (2^n-1) chips initially placed at the root. We prove a sorting property of terminal configurations of the process. We also analyze the end game moves poset and prove that this poset is a modular lattice.

我们研究了二叉树上的带标记芯片烧制过程,该过程的起点是最初放置在树根处的(2^n-1)个芯片。我们证明了该过程终端配置的排序属性。我们还分析了终局移动的位置集,并证明这个位置集是一个模态网格。
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引用次数: 0
The Hamilton Compression of Highly Symmetric Graphs 高度对称图的汉密尔顿压缩
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2023-12-13 DOI: 10.1007/s00026-023-00674-y
Petr Gregor, Arturo Merino, Torsten Mütze

We say that a Hamilton cycle (C=(x_1,ldots ,x_n)) in a graph G is k-symmetric, if the mapping (x_imapsto x_{i+n/k}) for all (i=1,ldots ,n), where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices (x_1,ldots ,x_n) equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a (360^circ /k) wedge of the drawing. The maximum k for which there exists a k-symmetric Hamilton cycle in G is referred to as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.

如果对于所有 (i=1,ldots,n)(这里的索引都是以 n 为模的),映射 (x_imapstox_{i+n/k})是 G 的自动变形,那么我们就说图 G 中的汉密尔顿循环 (C=(x_1,ldots ,x_n))是 k 对称的。换句话说,如果我们把顶点 (x_1,ldots ,x_n) 等距地画在一个圆上,并把 G 的边画成直线,那么 G 的画法就具有 k 倍旋转对称性,也就是说、关于图形的所有信息都被压缩到了绘图的一个 (360^circ /k)楔形中。我们研究了四个不同顶点变换图族的汉密尔顿压缩,它们分别是超立方体图、约翰逊图、永恒面图和无性群的卡莱图。在几种情况下,我们精确地确定了它们的汉密尔顿压缩率,在其他情况下,我们提供了接近的下限和上限。所构建的循环比文献中已知的几种经典格雷码的压缩率要高得多。我们的构造还产生了位串、组合和排列的灰色代码,这些代码的轨迹很少,而且/或者是平衡的。
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引用次数: 0
The MacMahon q-Catalan is Convex MacMahon q-Catalan 是凸的
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2023-12-12 DOI: 10.1007/s00026-023-00677-9
Tewodros Amdeberhan, Stephan Wagner

Let (nge 2) be an integer. We prove the convexity of the so-called MacMahon q-Catalan polynomials (C_n(q)=frac{1}{[n+1]_q}left[ 2n atop n right] _q) viewed as functions of q over the entire set of reals. Along the way, several auxiliary properties of the q-Catalan polynomials and intermediate results in the form of inequalities are presented, with the aim to make the paper self-contained. We also include a commentary on the convexity of the generating function for the integer partitions.

让 (nge 2) 是一个整数。我们证明了所谓的 MacMahon q-Catalan 多项式 (C_n(q)=frac{1}{[n+1]_q}left[ 2n atop n right] _q) 作为 q 在整个实数集上的函数的凸性。在此过程中,我们以不等式的形式介绍了 q-Catalan 多项式的几个辅助性质和中间结果,目的是使本文自成体系。我们还对整数分部生成函数的凸性进行了评述。
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引用次数: 0
期刊
Annals of Combinatorics
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