设m，n为正整数。将T（m，n）定义为由m × n个非负实矩阵组成的运输多面体，这些矩阵的行和为1，列和为m/n。特殊情况B（n）=T（n，n）是双随机阵的Birkhoff-von Neumann多面体。使用最近的非负整数矩阵的渐近计数（Canfield and McKay，2007），我们确定了当n趋于无穷大时T（m，n）的渐近体积，其中m=m（n）使得m/n既不减小也不增加太快。特别地，我们给出了B（n）的体积的一个渐近公式.
英文摘要：
Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m x n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case B(n)=T(n,n) is the much-studied Birkhoff-von Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determine the asymptotic volume of T(m,n) as n goes to infinity, with m=m(n) such that m/n neither decreases nor increases too quickly. In particular, we give an asymptotic formula for the volume of B(n).
We compute the set of facets of the polytope which is the convex hull of the Coxeter groups $\mathsf{F}_4$ or $\mathsf{H}_4$: For the group $\mathsf{F}_4$ we found $2$ orbits of facets which contradicts previous results published in \cite{birkhoff}. For the group $\mathsf{H}_4$ we found $1063$ orbits of facets which provides a counterexample to the conjecture of \cite{birkhoff}.
The Birkhoff polytope, defined to be the convex hull of $n\times n$ permutation matrices, is a well studied polytope in the context of the Ehrhart theory. This polytope is known to have many desirable properties, such as the Gorenstein property and existence of regular, unimodular triangulations. In this paper, we study analogues of the Birkhoff polytope for finite irreducible Coxeter groups of other types. We focus on a type-$B$ Birkhoff polytope $\mathcal{BB}(n)$ arising from signed permutation matrices and prove that it and its dual polytope are reflexive, and hence Gorenstein, and also pos…
Institute for Mathematics, Paderborn University（帕德博恩大学数学研究所）；Department of Mathematics and Computer Science, University of Puget Sound（普吉特湾大学数学与计算机科学系）；Department of Mathematics and Statistics, University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校数学与统计学系）；School of Mathematical and Statistical Sciences, Arizona State University（亚利桑那州立大学数理科学学院）
本文证明 B 型 c-Birkhoff 多胞体与堆偏序集的序多胞体单模等价，从而回答了此前关于 A 型结果的推广问题。
University of California, Riverside（加州大学河滨分校）；University of Puget Sound（普吉特海峡大学）；University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）；Arizona State University（亚利桑那州立大学）
The independence number of the Birkhoff polytope graph, and applications to maximally recoverable codes
最大可恢复代码是为分布式存储设计的代码，它结合了从单个节点故障中的快速恢复和从灾难性故障中的最佳恢复。Gopalan et al [SODA 2017]研究了网格拓扑中此类码所需的字母表大小，并给出了其组合特征。考虑完全二部图$K_{n,n}$的边的标签来自$F_2^d$，满足以下条件：对于任何简单圈，其边上的标签之和不为零。这是可能的最小d控制在n × n网格拓扑中最大可恢复代码所需的字母表大小。在当前工作之前，已知d在$(\log n)^2$和$n\log n$之间。我们改进了这两个界限，并证明了d在n中是线性的。上界是一个递归构造，它优于随机构造。下界如下首先有关的问题的独立数的Birkhoff多面体图，然后提供严格的界限，它使用的对称群的表示论。
Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces
Wydział Fizyki, Uniwersytet Warszawski（华沙大学物理系）；Centrum Fizyki Teoretycznej PAN（波兰科学院理论物理中心）；Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagielloński（雅盖隆大学斯莫卢霍夫斯基物理研究所）
Exact Ehrhart Series of Birkhoff Polytopes via Constant Terms and Finite-Field Evaluation
School of Mathematics and Statistics, Hainan University（海南大学数学与统计学院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）；Center for Combinatorics, LPMC, Nankai University（南开大学组合数学中心）
本文提出一种利用常数项和有限域精确计算 Birkhoff 多胞形 Ehrhart 级数的方法，通过根过滤器、对称函数和广义 Todd 系数实现高效计算，并推广到 World Cup 问题，给出直至阶数 12 的完整级数。
Institute of Computing, UFMS（乌法联邦大学计算研究所）；Faculty of Mathematics, University of Vienna（维也纳大学数学系）；School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）
We present a new algorithm for computing the volume of an arbitrary matroid base polytope. We provide two applications of this approach: a relation between the volume of the base polytope of a matroid $\M$ and its relaxation $\M'$, and a formula for the volume of an arbitrary sparse paving matroid base polytope.
We survey the computation of polytope volumes by the algorithms of Normaliz to which the Lawrence algorithm has recently been added. It has enabled us to master volume computations for polytopes from social choice in dimension $119$. This challenge required a sophisticated implementation of the Lawrence algorithm.
A complete classification of the $δ$-vectors of lattice polytopes whose normalized volumes are at most $4$ is known. In the present paper, we will classify all the $δ$-vectors of lattice polytopes with normalized volumes $5$.
The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.
In the frame of a classification of general square systems of polynomial equations solvable by radicals, Esterov and Gusev succeeded in classifying all spanning lattice polytopes whose normalized volumes are at most $4$. In the present paper, we complete to classify all lattice polytopes whose normalized volumes are at most $4$ based on the known classification of their $δ$-polynomials.
Let $(P,\leq_P)$ and $(Q,\leq_Q)$ be finite partially ordered sets with $|P|=|Q|=d$, and $\mathcal{C}(P) \subset \mathbb{R}^d$ and $\mathcal{C}(Q) \subset \mathbb{R}^d$ their chain polytopes. The twinned chain polytope of $P$ and $Q$ is the lattice polytope $Γ(\mathcal{C}(P),\mathcal{C}(Q)) \subset \mathbb{R}^d$ which is the convex hull of $\mathcal{C}(P) \cup (-\mathcal{C}(Q))$. It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give t…
We consider a minimum enclosing and maximum inscribed tropical balls for any given tropical polytope over the tropical projective torus in terms of the tropical metric with the max-plus algebra. We show that we can obtain such tropical balls via linear programming. Then we apply minimum enclosing and maximum inscribed tropical balls of any given tropical polytope to estimate the volume of and sample uniformly from the tropical polytope.
In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type A.
We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of Baldoni and Vergne's generalization of a volume formula originally due to Lidskii. We recover known flow polytope volume formulas and prove new volume formulas for flow polytopes that were seemingly unapproachable. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph. As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan nu…
Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinemen…
Recently, Benedetti et al. introduced an Ehrhart-like polynomial associated to a graph. This polynomial is defined as the volume of a certain flow polytope related to a graph and has the property that the leading coefficient is the volume of the flow polytope of the original graph with net flow vector $(1,1,\dots,1)$. Benedetti et al. conjectured a formula for the Ehrhart-like polynomial of what they call a caracol graph. In this paper their conjecture is proved using constant term identities, labeled Dyck paths, and a cyclic lemma.
The Lidskii formula for the type $A_n$ root system expresses the volume and Ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of Kostant partition functions. For every integer polytope the volume is the leading coefficient of the Ehrhart polynomial. The beauty of the Lidskii formula is the revelation that for these polytopes its Ehrhart polynomial function can be deduced from its volume function! Baldoni and Vergne generalized Lidskii's result for flow polytopes of arbitrary graphs $G$ and nonnegative integer netflows. While their formulas…
Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c$ that to the interior of $\mathcal{P}$. It follows from a lower bound theorem of Ehrhart polynomials that, when $c > 0$, the volume of $\mathcal{P}$ is bigger than or equal to $(dc + (d-1)b - d^2 + 2)/d!$. In the present paper, via triangulations, a short and elementary proof of the minimal volume formula is given.
The Baldoni--Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function fundamental in representation theory. On the other hand the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an integer polytope is its volume! Baldoni and Vergne proved these formulas via residues. To reveal the geometry of these f…
A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends several existing ones, and has been used to correct mistakes in other classifications. When specialized to three-dimensional s…
Adjacency polytopes appear naturally in the study of nonlinear emergent phenomena in complex networks. The "PQ-type" adjacency polytope, denoted $\nabla^{\mathrm{PQ}}_G$ and which is the focus of this work, encodes rich combinatorial information about power-flow solutions in sparse power networks that are studied in electric engineering. Of particular importance is the normalized volume of such an adjacency polytope, which provides an upper bound on the number of distinct power-flow solutions. In this article we show that the problem of computing normalized volumes for $\nabla^{\mathrm{PQ}}_G$…
