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The plethystic inverse of the odd Lie representations 奇数李表示的复数逆
Pub Date : 2020-03-24 DOI: 10.1090/proc/15938
S. Sundaram
The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $sum_{ngeq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations.
的Frobenius特征 $Lie_n,$ 对称群的表示 $S_n$ 由自由李代数提供,已知满足许多有趣的多面体恒等式。本文证明了Richard Stanley的一个关于和的多角形逆的猜想 $sum_{ngeq 0} Lie_{2n+1}$ 奇怪的谎言特征。的正则表示得到了一个明显新的多面体分解 $S_n$ 用钩子索引的不可约物,以及Lie表示。
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引用次数: 1
Determinant Identities for Toeplitz-Hessenberg Matrices with Tribonacci Number Entries 具有Tribonacci数项的Toeplitz-Hessenberg矩阵的行列式恒等式
Pub Date : 2020-03-24 DOI: 10.22108/TOC.2020.116257.1631
T. Goy, M. Shattuck
In this paper, we evaluate determinants of some families of Toeplitz-Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz-Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying an r-th order recurrence of a more general form with the appropriate initial conditions, where r>2 is arbitrary.
本文讨论了具有三波那契数项的一些Toeplitz-Hessenberg矩阵族的行列式。这些行列式公式也可以等价地表示为包含多项系数和三角波那契数乘积和的恒等式。特别是,我们通过Toeplitz-Hessenberg行列式建立了tribonacci与Fibonacci和Padovan序列之间的联系。然后,通过组合论证,我们得到了我们的行列式公式的广义tribonacci序列的扩展,它满足具有适当初始条件的更一般形式的r阶递推式,其中r>2是任意的。
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引用次数: 5
Introduction to dominated edge chromatic number of a graph 介绍图的支配边色数
Pub Date : 2020-03-19 DOI: 10.7494/OPMATH.2021.41.2.245
Mohammad R. Piri, S. Alikhani
We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$ is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by $chi_{dom}^{prime}(G)$. We obtain some properties of $chi_{dom}^{prime}(G)$ and compute it for specific graphs. Also we examine the effects on $chi_{dom}^{prime}(G)$ when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivision of $G$ and study the dominated edge chromatic number of these kind of graphs.
引入并研究了图的支配边着色。图$G$的支配边着色是$G$的适当边着色,使得每个颜色类都被$G$的至少一条边支配。所有主导边着色的最小颜色数称为主导边色数,用$chi_{dom}^{prime}(G)$表示。我们得到了$chi_{dom}^{prime}(G)$的一些性质,并对特定图进行了计算。我们还研究了当$G$被对$G$的顶点和边的操作所修改时,对$chi_{dom}^{prime}(G)$的影响。最后,我们考虑了$G$的$k$细分,并研究了这类图的主导边色数。
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引用次数: 0
Restricted color $n$-color compositions 限制颜色$n$-颜色组合
Pub Date : 2020-03-11 DOI: 10.4310/JOC.2021.v12.n2.a8
Brian Hopkins, Hua Wang Saint Peter's University, Georgia Southern University
Agarwal introduced $n$-color compositions in 2000 and most subsequent research has focused on restricting which parts are allowed. Here we focus instead on restricting allowed colors. After three general results, giving recurrence formulas for the cases of given allowed colors, given prohibited colors, and colors satisfying modular conditions, we consider several more specific conditions, establishing direct formulas and connections to other combinatorial objects. Proofs are combinatorial, mostly using the notion of spotted tilings introduced by the first named author in 2012.
Agarwal在2000年引入了n种颜色的组合,随后的大多数研究都集中在限制哪些部分是允许的。这里我们关注的是限制允许的颜色。在三个一般结果之后,给出了给定允许颜色、给定禁止颜色和满足模条件的颜色的递推公式,我们考虑了几个更具体的条件,建立了直接公式和与其他组合对象的联系。证明是组合的,主要使用2012年第一位作者引入的斑点拼接的概念。
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引用次数: 0
Unitary signings and induced subgraphs of Cayley graphs of $mathbb{Z}_2^{n}$ $mathbb{Z}_2^{n}$的Cayley图的幺正符号与诱导子图
Pub Date : 2020-03-10 DOI: 10.19086/AIC.17912
N. Alon, Kai Zheng
Let $G$ be a Cayley graph of the elementary abelian $2$-group $mathbb{Z}_2^{n}$ with respect to a set $S$ of size $d$. We prove that for any such $G, S$ and $d$, the maximum degree of any induced subgraph of $G$ on any set of more than half the vertices is at least $sqrt d$. This is deduced from the recent breakthrough result of Huang who proved the above for the $n$-hypercube $Q^n$, in which the set of generators $S$ is the set of all vectors of Hamming weight $1$. Motivated by his method we define and study unitary signings of adjacency matrices of graphs, and compare them to the orthogonal signings of Huang. As a byproduct, we answer a recent question of Belardo et. al. about the spectrum of signed $5$-regular graphs.
设$G$是初等阿贝尔$2$-群$mathbb{Z}_2^{n}$关于大小为$d$的集合$S$的Cayley图。我们证明了对于任意这样的$G, $ S$和$d$, $G$的任何诱导子图在任何超过一半顶点的集合上的最大度至少为$根号d$。这是从Huang最近的突破性成果中推导出来的,他证明了$n$-超立方体$Q^n$,其中的生成元集$S$是Hamming权值$1$的所有向量的集合。在他的方法的启发下,我们定义并研究了图的邻接矩阵的酉符号,并将其与黄的正交符号进行了比较。作为一个副产品,我们回答了Belardo等人最近关于$5$正则图的谱的问题。
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引用次数: 3
Determinantal formulas for dual Grothendieck polynomials 对偶格罗滕迪克多项式的行列式公式
Pub Date : 2020-03-09 DOI: 10.1090/proc/16008
A. Amanov, Damir Yeliussizov
We prove Jacobi-Trudi-type determinantal formulas for skew dual Grothendieck polynomials which are $K$-theoretic deformations of Schur polynomials. We also prove a bialternant type formula analogous to the classical definition of Schur polynomials.
证明了Schur多项式的K -理论变形的斜对偶Grothendieck多项式的jacobi - trudi型行列式。我们还证明了一个类似于经典舒尔多项式定义的双交替型公式。
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引用次数: 11
Higher discrete homotopy groups of graphs 图的高离散同伦群
Pub Date : 2020-03-05 DOI: 10.5802/ALCO.151
Bob Lutz
This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if $G$ is a graph containing no 3- or 4-cycles, then the $n$th discrete homotopy group $A_n(G)$ is trivial for all $ngeq 2$. Second we exhibit for each $ngeq 1$ a natural homomorphism $psi:A_n(G)to mathcal{H}_n(G)$, where $mathcal{H}_n(G)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which $mathcal{H}_n(G)$ is nontrivial and $psi$ is surjective. It follows that for each $ngeq 1$ there are graphs $G$ for which $A_n(G)$ is nontrivial.
本文研究了Barcelo等人引入的图的离散同伦理论。我们证明了两个主要结果。首先,我们证明了如果$G$是一个不包含3圈或4圈的图,那么$n$第1个离散同伦群$A_n(G)$对于所有$ngeq 2$都是平凡的。其次,我们为每个$ngeq 1$展示了一个自然同态$psi:A_n(G)to mathcal{H}_n(G)$,其中$mathcal{H}_n(G)$是$n$第一个离散三次奇异同态群,以及一个无限族的图$G$,其中$mathcal{H}_n(G)$是非平凡的,$psi$是满射的。由此可见,对于每个$ngeq 1$,都有一些图形$G$,其中$A_n(G)$是非平凡的。
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引用次数: 4
NON‐EULERIAN DEHN–SOMMERVILLE RELATIONS 非欧拉DEHN - SOMMERVILLE关系
Pub Date : 2020-02-29 DOI: 10.1112/MTK.12072
Connor Sawaske, Lei Xue
The classical Dehn--Sommerville relations assert that the $h$-vector of an Eulerian simplicial complex is symmetric. We establish three generalizations of the Dehn--Sommerville relations: one for the $h$-vectors of pure simplicial complexes, another one for the flag $h$-vectors of balanced simplicial complexes and graded posets, and yet another one for the toric $h$-vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of "errors coming from the links." For simplicial complexes, this further extends Klee's semi-Eulerian relations.
经典的Dehn—Sommerville关系断言欧拉简单复合体的h向量是对称的。我们建立了Dehn—Sommerville关系的三种推广:一种是关于纯单纯复形的$h$-向量,另一种是关于平衡单纯复形和梯度序集的标志$h$-向量,还有一种是关于具有限制奇点的梯度序集的环$h$-向量。在所有这些情况下,我们用“来自链接的误差”来表达任何对称性的失败。对于简单复合体,这进一步扩展了Klee的半欧拉关系。
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引用次数: 2
Zeta functions of periodic cubical lattices and cyclotomic-like polynomials. 周期立方格的ζ函数与类环形多项式。
Pub Date : 2020-02-27 DOI: 10.2969/aspm/08410093
Y. Hiraoka, Hiroyuki Ochiai, T. Shirai
Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.
通过计算邻接算子及其特征多项式的所有特征值,显式导出了周期三次格的ζ函数。我们引入类环多项式来给出zeta函数的因式分解,并计算与每个类环多项式相关的伽罗瓦作用的轨道数以得到其进一步的因式分解。给出了多项式不可约的充分必要条件,并由此讨论了多项式的不可约性。
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引用次数: 0
Dual Grothendieck polynomials via last-passage percolation 末道渗流的对偶格罗滕迪克多项式
Pub Date : 2020-02-24 DOI: 10.5802/crmath.67
Damir Yeliussizov
The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.
对称函数环的基是对偶格罗登狄克多项式,它是舒尔多项式的非齐次K理论变形。我们证明了对偶格罗滕迪克多项式决定了有向最后通道渗流模型的柱分布。
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引用次数: 8
期刊
arXiv: Combinatorics
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