On multi-symmetric functions and transportation polytopes

Tropical determinant on transportation polytope

Let ${\mathcal D}^{k,l}(m,n)$ be the set of all the integer points in the transportation polytope of $kn\times ln$ matrices with row sums $lm$ and column sums $km$. In this paper we find the sharp lower bound on the tropical determinant over the set ${\mathcal D}^{k,l}(m,n)$. This integer piecewise-linear programming problem in arbitrary dimension turns out to be equivalent to an integer non-linear (in fact, quadratic) optimization problem in dimension two. We also compute the sharp upper bound on a modification of the tropical determinant, where the maximum over all the transversals in a matr…

Orthogonal bases for transportation polytopes applied to Latin squares, magic squares and Sudoku boards

We give a simple construction of an orthogonal basis for the space of m by n matrices with row and column sums equal to zero. This vector space corresponds to the affine space naturally associated with the Birkhoff polytope, contingency tables and Latin squares. We also provide orthogonal bases for the spaces underlying magic squares and Sudoku boards. Our construction combines the outer (i.e., tensor or dyadic) product on vectors with certain rooted, vector-labeled, binary trees. Our bases naturally respect the decomposition of a vector space into centrosymmetric and skew-centrosymmetric piec…

运输多面体的Groebner基

Groebner Bases for Transportation Polytopes

$3\times 3$运输多面体$T$的复曲面理想是二次生成的。唯一的例外是Birkhoff多面体$B_3$。如果$T$不是$B_3$的倍数，这些理想甚至有无平方二次初始理想.此类包含所有光滑$3 \times 3$传输多面体。

Counting Integer Points in Multi-Index Transportation Polytopes

We expand on a result of Barvinok and Hartigan to derive asymptotic formulas for the number of integer and binary integer points in a wide class of multi-index $k_1\times k_2\times \ldots \times k_ν$ transportation polytopes. A simple closed form approximation is given as the $k_j$s go to infinity.

The Hierarchy of Circuit Diameters and Transportation Polytopes

The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still open whether the Hirsch conjecture is true for general $m{\times}n$--transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. The key reason was that this hierarchy provides some interesting lower bounds for the usual graph diameter. This paper has three contributions: First, we compare the hierarc…

Information-Geometric Equivalence of Transportation Polytopes

This paper deals with transportation polytopes in the probability simplex (that is, sets of categorical bivariate probability distributions with prescribed marginals). Information projections between such polytopes are studied, and a sufficient condition is described under which these mappings are homeomorphisms.

图与随机演化树上的最优传输

Optimal Transport on Graphs and Stochastically Evolving Trees

该研究针对图上概率密度函数的最优传输问题，通过分析关联多面体提出最速下降算法，可高效计算1-Wasserstein距离，并将其应用于图的Ollivier-Ricci曲率计算。

A tale of centrally symmetric polytopes and spheres

This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the upper bound theorem for centrally symmetric simplicial spheres to the generalized lower bound theorem for centrally symmetric simplicial polytopes and the lower bound conjecture for centrally symmetric simplicial spheres and manifolds.

A lower bound theorem for centrally symmetric simplicial polytopes

Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\geq 3$, $g_2(P) \geq {d \choose 2}-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\geq 4$ that satisfy this inequality as equality. This gives a natural generalization of the classical Lower Bound Theorem for simplicial polytopes to the setting of centrally symmetric simplicial polytopes.

Transportation Polytope and its Applications in Parallel Server Systems

A parallel server system is a stochastic processing network with applications in manufacturing, supply chain, ride-hailing, call centers, etc. Heterogeneous customers arrive in the system, and only a subset of servers can serve any customer type given by the flexibility graph. The goal of the system operator is to minimize the delay that depends on the scheduling policy and the flexibility graph. A long line of literature focuses on designing near-optimal scheduling policies given a flexibility graph. On the contrary, we fix the scheduling policy to be the so-called MaxWeight scheduling given …

运输多面体的列联表数、整数流和体积的渐近估计

Asymptotic estimates for the number of contingency tables, integer flows, and volumes of transportation polytopes

我们证明了一个渐近估计的数量mxn非负整数矩阵（列联表）与规定的行和列和，更一般地说，在网络中的整数可行流的数量。类似地，我们估计了具有指定行和列和的mxn非负实矩阵的多面体的体积。我们的估计是凸优化问题的解决方案，因此可以有效地计算。作为推论，我们证明了如果行和R=（r_1，.，r_m）和列和C=（c_1，...，c_n），其中r_1 + ... + r_m =c_1 +. +c_n =N离常向量足够远，则在元素总和为N的m × n个非负整数矩阵的一致概率空间中，行和为R的矩阵所组成的事件与列和为C的矩阵所组成的事件渐近正相关。

Ehrhart theory of symmetric edge polytopes via ribbon structures

Using a ribbon structure of the graph, we construct a dissection of the symmetric edge polytope of a graph into unimodular simplices. Our dissection is shellable, and one can interpret the elements of the resulting $h$-vector via graph theory. This gives an elementary method for computing the $h^*$-vector of the symmetric edge polytope.

对称边多面体的酉作用与等变体积

Unitary actions and equivariant volumes of symmetric edge polytopes

该研究刻画了对称边多面体的刚性对称性，证明其酉等价等价于对应图同构，建立了对称群作用下不动子集的体积与小图对称边多面体的关系，并给出完全图情形的特例结果。

Facets and facet subgraphs of symmetric edge polytopes

Symmetric edge polytopes, a.k.a. PV-type adjacency polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the study of the algebraic Kuramoto equations of unmixed form whose Newton polytopes are the symmetric edge polytopes. The interplay between the geometric structure of symmetric edge polytopes and the topological structure of the underlying graphs has been a recurring theme in recent studies. In particular, ``facet/face su…

凸多胞体的移位对称$δ$ -向量

Shifted symmetric $δ$-vectors of convex polytopes

如果对于每个$0 \leq i \leq [(d-1)/2]$的$δ_{d-i} = δ_{i+1}$，则称为$δ$ -向量$δ(\Pc)= (δ_0, δ_1, ..., δ_d)$移位对称.将研究具有移位对称$δ$ -向量的$(0,1)$ -多面体的自然家族。

一个中心对称的多面体能有多邻近？

How neighborly can a centrally symmetric polytope be?

证明了存在k-邻中心对称的2（n+d）阶d维多面体，其中k（d，n）=Theta（d/（1+log（（d+n）/d）.我们还证明了这个界是紧的。

Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices

In Ehrhart theory, the $h^*$-vector of a rational polytope often provide insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal $h^*$-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this paper, we examine the $h^*$-vectors of a class of polytopes containing real doubly-stochastic symmetric matrices.

From acute sets to centrally symmetric $2$-neighborly polytopes

What is the maximum number of vertices that a centrally symmetric 2-neighborly polytope of dimension $d$ can have? It is known that the answer does not exceed $2^d$. Here we provide an explicit construction showing that it is at least $2^{d-1}+2$.

对称边多胞体不是γ-正的

Symmetric edge polytopes are not gamma-positive

通过构造无穷反例族，否证了Ohsugi和Tsuchiya关于对称边多胞体的Ehrhart h^*-多项式是γ-正的猜想，最小反例为36维对称边多胞体。

Newton polytopes and symmetric Grothendieck polynomials

Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial $K$-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case.

2-划分极大对称多面体的 IDP

IDP for 2-Partition Maximal Symmetric Polytopes

本文提出证明对称多面体整数分解性质的框架，并据此证明位于 $\mathbb{R}^3$ 超平面中的 $2$-划分极大多面体满足 IDP，关键在于证明特定多项式集合具有饱和 Newton 多面体。

源于联图的对称边多面体的面数

Number of facets of symmetric edge polytopes arising from join graphs

本文研究了连通图对称边多面体面数的上下界猜想，证明了该猜想对任意两图的联图成立，从而为 Nill 关于反身多面体面数上界的猜想提供了部分回答。

伪对称光滑Fano多胞体极对偶的魔幻正性

Magic positivity for polar duals of pseudo-symmetric smooth Fano polytopes

本文证明了伪对称单纯反射多胞体的极对偶的Ehrhart多项式具有魔幻正性，并进一步证明伪对称光滑Fano多胞体的魔幻系数严格为正，从而得出其极对偶Ehrhart正且$h^*$-多项式实根且$\u03b3$-正，同时解决了Konoike关于循环对称边多胞体的问题。

对称凹多边形

Symmetric Alcoved Polytopes

广义凹多面体是指多面体的面法线是给定根系的根。如果不存在包含它的严格更小的凹穴多面体，我们称凹穴多面体中的点集为生成集。$A$型凹穴多面体正是通常意义上也是凸的热带多面体。在这种情况下，热带发电机形成发电机组。我们表明，对于任何根系以外的$F_4$，每一个凹多面体下的自然外尔群的作用下不变的基数等于考克斯特数的根系的生成集。

对称边多面体的边数

The number of edges of a symmetric edge polytope

研究简单图的对称边多面体边数，给出基于基本图不变量的尖锐下界并刻画达到此界的图，还探讨与h*多项式的联系及边删除时该多项式的行为。

随机对称边多面体的面、度序列与聚类

Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

研究随机图模型下对称边多面体的面结构，发现Erdős-Renyi图的阈值概率，并证明固定度序列下平均局部聚类系数与面数量正相关。

少边图的对称边多胞体的面数

Facets of Symmetric Edge Polytopes for Graphs with Few Edges

本文研究少边连通图的对称边多胞体，引入最大与最小面数序列，建立稀疏图类的面数公式，并推进识别面数最大化图的猜想。

非中心对称反射多面体的面数：来自偏序集的双链多面体

Facet numbers of non-centrally symmetric reflexive polytopes arising from posets

本文研究了双链多面体的面数，证明其至多为6^{d/2}，并为Nill猜想提供了部分答案。

关于对称边多面体的γ向量

On the gamma-vector of symmetric edge polytopes

该研究从确定性与概率性双视角研究对称边多面体的γ向量，证明了任意图γ₂的非负性及零值刻画，验证了相关猜想，并证实随机图情形下Gal猜想渐近几乎必然成立。

Lattice polytopes from Schur and symmetric Grothendieck polynomials

Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove tha…

关于完全多部图的对称边多胞体Ehrhart多项式根的一个猜想

On a Conjecture Concerning the Roots of Ehrhart Polynomials of Symmetric Edge Polytopes from Complete Multipartite Graphs

本文针对完全多部图对称边多胞体Ehrhart多项式根的猜想，沿用已有方法验证了若干新类，并指出原方法可能不足以证明一般情形。

对称边多面体的闵可夫斯基可分解性

Minkowski decomposability of symmetric edge polytopes

该研究刻画了有限简单图的对称边多面体为闵可夫斯基可分解的充要条件，证明仅当图为三类完全多部图时满足该性质，其余情况不可分解。

The asymptotic volume of diagonal subpolytopes of symmetric stochastic matrices

The asymptotic volume of the polytope of symmetric stochastic matrices can be determined by asymptotic enumeration techniques as in the case of the Birkhoff polytope. These methods can be extended to polytopes of symmetric stochastic matrices with given diagonal, if this diagonal varies not too wildly. To this end, the asymptotic number of symmetric matrices with natural entries, zero diagonal and varying row sums is determined and a third order correction factor to this is examined.

关于广义对称边多面体的Ehrhart理论

On the Ehrhart Theory of Generalized Symmetric Edge Polytopes

本文通过组合学和格基技术，将SEPs的性质扩展到广义SEPs，并证明广义SEPs不一定是γ非负的，但通过删除两个元素可得到γ非负的SEPs，支持Ohsugi和Tsuchiya的猜想。

Symmetric, Hankel-symmetric, and Centrosymmetric Doubly Stochastic Matrices

We investigate convex polytopes of doubly stochastic matrices having special structures: symmetric, Hankel symmetric, centrosymmetric, and both symmetric and Hankel symmetric. We determine dimensions of these polytopes and classify their extreme points. We also determine a basis of the real vector spaces generated by permutation matrices with these special structures.

伪对称单纯自反多面体的分类

Classification of pseudo-symmetric simplicial reflexive polytopes

一个自反多胞形，或其相应的Gorenstein复曲面Fano簇，称为伪对称的，如果多胞形有一对中心对称的小平面。本文给出了伪对称单纯自反多面体的一个完整分类及其应用。这推广了Ewald关于伪对称非奇异环面Fano簇的一个结果。

多胞形的平移镶嵌，具有多重性

Translational tilings by a polytope, with multiplicity

研究了凸体P的平移重叠覆盖R^d的问题，使得R^d中几乎每一点都被覆盖k次.这样的平移覆盖欧几里得空间称为k-平铺。通过翻译对镶嵌（即本文中的1-镶嵌）的研究始于费多罗夫和闵可夫斯基的工作。在这里，我们通过证明如果一个凸体通过平移使R^d k-平铺，则它是中心对称的，并且它的刻面也是中心对称的，将闵可夫斯基的研究扩展到k-平铺。这些是类似于闵可夫斯基条件的1-平铺多面体。反过来，在P是有理多面体的情况下，我们也证明了：如果P是中心对称的，并且有中心对称的小平面，那么P必k-tile R^d，其中有某个正整数k。

Many faces of symmetric edge polytopes

Symmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization model in physics -- where they are called adjacency polytopes -- and to Kantorovich--Rubinstein polytopes from finite metric space theory. Each of these connections motivates the study of symmetric edge polytopes of particular classes of graphs. We focus on such classes and apply algebraic-combinatorial methods to investigate invariants of the associated symmetric edge polytopes.

Reconstruction of symmetric convex bodies from Ehrhart-like data

In a previous paper, we showed how to use the Ehrhart function $L_P(s)$, defined by $L_P(s) = \#(sP \cap \mathbb Z^d)$, to reconstruct a polytope $P$. More specifically, we showed that, for rational polytopes $P$ and $Q$, if $L_{P + w}(s) = L_{Q + w}(s)$ for all integer vectors $w$, then $P = Q$. In this paper we show the same result, but assuming that $P$ and $Q$ are symmetric convex bodies instead of rational polytopes.

全对称自补平面分拆矩阵及相关多胞体

Totally symmetric self-complementary plane partition matrices and related polytopes

本文建立从全对称自补平面分拆到magog矩阵的双射，研究其计数性质，并分析相关多胞体的极点、不等式、顶点与面。

对称（co）同调多面体

Symmetric (co)homology polytopes

本文提出了一种将单纯复形的拓扑性质转化为对称多面体几何性质的方法，并研究了其整数分解性质及反射性。

分裂图的中心对称多面体的面数

Face numbers of centrally symmetric polytopes from split graphs

我们分析了一类引人注目的中心对称多面体，分裂图的汉森多面体。我们证实了Kalai对这类多面体（它们都至少有3^d非空面）的3^d猜想，并证明了其中的Hanner多面体（恰好有3^d非空面）对应于阈值图。我们的研究产生了一个新的汉森多面体家族，只有3^d+16个非空面。

关于对称边多胞体向正则拟阵的推广

On a generalization of symmetric edge polytopes to regular matroids

本文将对称边多胞体推广到正则拟阵，证明其幺模等价性由图形拟阵决定，并给出广义多胞体的面描述、显式正则幺模三角剖分及其极多胞体 Ehrhart 理论与对偶正则拟阵流格的联系。

邻中心对称多面体的显式构造

An explicit construction for neighborly centrally symmetric polytopes

基于Hadamard矩阵，给出了具有4d顶点的中心对称d维多面体的无穷级数的显式构造.这似乎是最好的明确版本，但最近的概率结果，由于Linial和Novik，谁证明了存在这样的多面体与相邻的d/400。

多面体、霍普夫代数与拟对称函数

Polytopes, Hopf algebras and Quasi-symmetric functions

本文利用霍普夫代数和拟对称函数的技巧来研究组合多胞形。考虑由所有组合多面体生成的自由阿贝尔群$\mathcal{P}$。在这个群上有两个自然的双线性运算，它们由一个直积$\times $和一个多面体的连接$\divideontimes$定义。$(\mathcal{P},\times)$是交换结合双阶多项式环，$\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes)$是交换结合三阶多项式环.环$\mathcal{RP}$具有分次Hopf代数的结构。事实证明，$\mathcal{P}$在$\mathcal{RP}$上具有自然的Hopf余模结构。面操作符$d_k$发送一个多面体的总和，其所有$(n-k)$维面定义在两个环的霍普夫模结构的普遍莱布尼茨-霍普夫代数$\mathcal{Z}$。该结构给出环同态$\R\to\Qs\otimes\R$，其中$\R$是$\mathcal{P}$或$\mathcal{RP}$。将这个同态与$\mathcal{P}$的特征标$P^n\toα^n$，$\mathcal{RP}$的特征标$P^n\toα^{n+1}$，以及它的反同态合成，得到环同态$f\colon\mathcal{P}\to\Qs[α]$，$f_{\mathcal{RP}}\colon\mathcal{RP}\t…

中心对称多面体的广义h-向量的下界

Lower Bounds for the Generalized h-Vectors of Centrally Symmetric Polytopes

在前一篇文章中，我们证明了一个中心对称有理多面体的广义$h$ -向量的系数的紧下界使用相关的投射环面簇的交上同调。在这里，我们提出了一个新的证明的基础上的组合相交上同调理论的Barthel，Atelet，Fieseler和Kaup。这个理论也是有效的非理性多面体时，没有相应的环面品种。因此，我们可以建立中心对称多面体的界，甚至不要求它们是有理的。

Smooth centrally symmetric polytopes in dimension 3 are IDP

In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric $3$-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.

循环多胞形的中心对称形式

A centrally symmetric version of the cyclic polytope

我们定义了一个中心对称的类似物的循环多面体，并研究其面部结构。我们猜想，我们的多面体提供渐近的最大数量的面在所有维度之间的所有中心对称多面体与n个顶点的一个给定的偶数维d=2k时，d是固定的，n的增长。对偶数维d=2k和整数0< j <k，我们证明了n个顶点的中心对称d维多面体的最大可能j维面数至少是（cj（d）+o（1））{n \choose j+1}，当cj（d）>0时，最大可能是（1-2^{-d}+o（1））{n \choose j+1}.我们证明了c_1（d）\geq（d-2）/（d-1）.

色对称函数的牛顿多面体与洛伦兹性质

The Newton polytope and Lorentzian property of chromatic symmetric functions

本文研究Dyck路径无差别图的色对称函数，证明其允许着色权重构成置换多面体的格点并给出支配权公式，猜想并部分证明其洛伦兹性质，同时将牛顿多面体结果推广到(3+1)-自由偏序集。

多面中心对称多面体

Centrally symmetric polytopes with many faces

我们提出了明确的建设中心对称多面体与许多方面：首先，我们构造一个d维中心对称多面体P，（1.316）n d个顶点，使得P的每对非对映顶点跨越P的一条边，其次，对于整数k>1，我们构建了一个d-一个任意高维数d的中心对称多面体P，其顶点数N任意大，使得对于某个0 < delta_k < 1（1-delta_k^d）{N choose k}顶点集的k个子集跨越P的面，第三，对于整数k>1和a>0，我们构造一个中心对称多面体Q，它具有任意大的顶点数N和维数d=k^{1+o（1）}，使得最小（1 - k^{-a}）{N choose k}顶点集的k-子集跨越Q的面。

利用剖分生成树的活动性的图多面体$h^*$-向量

$h^*$-vectors of graph polytopes using activities of dissecting spanning trees

该研究针对图对称边多面体与半平衡有向图根多面体的生成树剖分，证明其$h^*$-多项式可表为对应生成树活动性的生成函数，提供了计算新方法并揭示了剖分统计量分布的无关性。

良对称多项式的牛顿多胞体

Newton polytope of good symmetric polynomials

本文提出一类一般的对称多项式，证明其牛顿多胞体具有饱和性和整数分解性质，并统一涵盖多种已有对称多项式。

高维格点多面体的中心极限定理：对称边多面体

Central limit theorems for high dimensional lattice polytopes: symmetric edge polytopes

本文研究了由Erdős--Rényi随机图生成的高维对称边多面体，推导了其边数和单模三角剖分边数的中心极限定理，并揭示了主导方差项抵消导致的非典型波动机制。

Integer points enumerator of hypergraphic polytopes

For a hypergraphic polytope there is a weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines its f-polynomial. This quasisymmetric function invariant of hypergraphs extends the Stanley chromatic symmetric function of simple graphs. We consider a certain combinatorial Hopf algebra of hypergraphs and show that universal morphism to quasisymmetric functions coincides with this enumerator function.