Luis Ferroni
In this paper we investigate the Ehrhart Theory of the independence matroid polytope of uniform matroids. It is proved that these polytopes have an Ehrhart polynomial with positive coefficients. To do that, we prove that indeed all half-open-hypersimplices are Ehrhart positive, and tile disjointly our polytope using them.
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The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two constructions of triangulations of hypersimplices due to Stanley and Sturmfels and explain them in terms of alcoved polytopes. We study triangulations of alcoved polytopes, the adjacency graphs of these triangulations, and give a combinatorial formula for volumes of these polytopes. In particular, we study a class of matroid polytopes, which we call the multi-hypersimp…
Below is a method for relating a mixed volume computation for polytopes sharing many facet directions to a symmetric random walk. The example of permutahedra and particularly hypersimplices is expanded upon.
We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that generalize descents and big ascents. For ranked posets, Lipschitz polytopes are centrally-symmetric and Gorenstein, which implies symmetry and unimodality of the statistics. Finally, we define $(P,k)$-hypersimplices as generalizations of classical hypersimplices and give combinatorial interpretations of their volumes and $h^*$-vectors.
An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.
We consider the Ehrhart $h^*$-vector for the hypersimplex. It is well-known that the sum of the $h_i^*$ is the normalized volume which equals an Eulerian numbers. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the $h^*_i$ coefficients in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.
Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular triangulation of the hypersimplex restricts to a triangulation of each r-stable hypersimplex. For the case of the second hypersimplex defined by the two-element subsets of an n-set, we provide a shelling of this triangulation that sequentially shells each r-stable sub-hypersimplex. In this case, we utilize the shelling to compute the Ehrhart h*-polynomials o…
In this paper we present an explicit combinatorial description of a special class of facets of the secondary polytopes of hypersimplices. These facets correspond to polytopal subdivisions called multi-splits. We show a relation between the cells in a multi-split of the hypersimplex and nested matroids. Moreover, we get a description of all multi-splits of a product of simplices. Additionally, we present a computational result to derive explicit lower bounds on the number of facets of secondary polytopes of hypersimplices.
Describing the combinatorial structure of the tropical complex $C$ of a tropical matroid polytope, we obtain a formula for the coarse types of the maximal cells of $C$. Due to the connection between tropical complexes and resolutions of monomial ideals, this yields the generators for the corresponding coarse type ideal introduced in a recent paper of Dochtermann, Joswig and Sanyal (2010, preprint arXiv.org:1001.0237). Furthermore, a complete description of the minimal tropical halfspaces of the uniform tropical matroid polytopes, i.e. the tropical hypersimplices, is given.
Lineup polytopes of product of simplices
本文将量子化学中阵容多胞体的不等式表示技术推广到量子信息论，研究单纯形乘积顶点的阵容多胞体，利用其组合与对称性给出算法解，并揭示其与Gale序、标准杨表及Resonance构型的联系。
The Ehrhart $h^*$-polynomials of positroid polytopes
该研究针对正广面多胞形的Ehrhart $h^*$-多项式展开，给出了基于排列下降的任意正广面多胞形$h^*$-多项式显式公式，推广了此前针对超单纯形的相关结果。
In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.
Ehrhart positivity for a certain class of panhandle matroids
该研究给出一类加权多超单形的Ehrhart系数组合公式，并证明柄状拟阵$\textrm{Pan}_{k,n-2,n}$基多胞形的Ehrhart系数具有正性。
We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstr…
Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including zonotopal decompositions, symmetric triangulations, combinatorial interpretation of the $h^\ast$-polynomial, and certificates for the (non)existence of invariant non-degenerate hypersurfaces. We apply these methods to several families of examples including hypersimplices, orbit polytopes, and graphic zonotopes, expanding the library of polytopes for which their …
Graded Ehrhart theory for hypersimplices
该研究证实了Reiner和Rhoades关于立方体超平面切片q-埃哈特级数的猜想，通过找到轨道调和理想生成集等方法，证明超单形的调和代数有限生成，给出其有理性的第二个证明。
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d-simplex. Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the d-step conjecture.
Weighted lattice point enumeration in lecture hall order polytopes
本文研究讲座厅序多胞形中的加权格点枚举，发现其半开多胞形对多项式权重封闭，并证明有根树的加权$h^\ast$-多项式实根，补充了先前结果。
We develop a procedure for the complete computational enumeration of lattice $3$-polytopes of width larger than one, up to any given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most eleven lattice points (there are 216,453 of them). In order to achieve this we prove that if $P$ is a lattice 3-polytope of width larger than one and with at least seven lattice points then it fits in one of three categories that we call boxed, spiked and merged. Boxed polytopes have at most 11 lattice points; in particular they are finitely many, and we enume…
Lattice point enumeration of polytopes associated to integer compositions
本文证明了Chapoton关于整数组合关联多面体h向量的猜想，并给出了gamma向量非负性及h*多项式组合解释。
Face enumeration for split matroid polytopes
本文显式研究了拟阵多面体的面计数，证明分裂拟阵多面体的面数仅取决于循环平坦的数量及模对信息，并提供无需取凸包或计算面格即可计算$f$-向量的公式。
Enumeration and constructions of vertices of the polytope of polystochastic matrices
本文针对多随机矩阵构成的伯克霍夫凸包Ωₙᵈ，修正前人结果确定Ω₄³和Ω₃⁴的所有顶点，提出基于多维矩阵乘积的顶点构造方法，并找到d≥4时Ω₃ᵈ中具大支撑的对称顶点。
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…
Generalized snake posets, order polytopes, and lattice-point enumeration
本文研究了广义蛇偏序集的顺序聚类的Ehrhart理论，推导了其$h^*$-多项式的显式公式和递推关系，并证明了其$h^*$-向量的界性。
A Facet Enumeration Algorithm for Convex Polytopes
本文提出一种简单新颖的凸多面体面枚举算法，在Matlab中实现，并通过已知多面体测试验证其有效性和效率，且因对偶性可推广至顶点枚举。
Lattice point enumeration of some arbor polytopes
本文研究了某些树形多面体的格点计数问题，通过新的组合模型证明了其Ehrhart多项式为魔术正，并推导了h*多项式的实根性。
We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. In addressing this problem, we study the {\em Shaped Partition Polytope} defined as the convex hull of solutions. The Shaped Partition Problem captures ${\cal N}{\cal P}$-hard problems such as the Max-Cut problem and the Traveling Salesperson problem, and the Shaped Partition Polytope may have exponentially ma…
Ehrhart Bounds for Panhandle and Paving Matroids Through Enumeration of Chain Forests
本文通过有序链森林的枚举，证明了Panhandle拟阵多面体的Ehrhart正性及铺砌拟阵的Ferroni系数上界猜想。
Polytopes and posets associated to preorders
本文引入并研究有限集上预序定义的预序多面体，从格点计数角度推广了树形多面体，证明了其作为格多面体的对偶性，给出了归一化体积的组合解释以及Ehrhart多项式和h*多项式的公式。
The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of th…
We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we completely enumerate all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 a…
Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e. non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial t…
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conjectures. In this paper, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids --- a combinatorial abstraction of neighborly polytopes --- of small rank and corank. In particular, if we denote by OM($r,…
We give a complete enumeration of all 2-neighborly 0/1-polytopes of dimension 7. There are 13 959 358 918 different 0/1-equivalence classes of such polytopes. They form 5 850 402 014 combinatorial classes and 1 274 089 different f-vectors. It enables us to list some of their combinatorial properties. In particular, we have found a 2-neighborly polytope with 14 vertices and 16 facets.
We report here a computation giving the complete list of facets for the cut polytopes over several very symmetric graphs with $15-30$ edges, including $K_8$, $K_{3,3,3}$, $K_{1,4,4}$, $K_{5,5}$, some other $K_{l,m}$, $K_{1,l,m}$, $Prism_7, APrism_6$, Möbius ladder $M_{14}$, Dodecahedron, Heawood and Petersen graphs. For $K_8$, it shows that the huge lists of facets of the cut polytope $CUTP_8$ and cut cone $CUT_8$, given in [CR] is complete. We also confirm the conjecture that any facet of $CUTP_8$ is adjacent to a triangle facet. The lists of facets for $K_{1,l,m}$ with $(l,m)=(4,4),(3,5),(3,…
A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Our approach is inductive: for each fixed $(d-1)$-dimensional $2$-level polytop…
The combinatorics of reduced words and commutation classes plays an important role in geometric representation theory. A string polytope is a lattice polytope associated to each reduced word of the longest element $w_0$ in the symmetric group which encodes the character of a certain irreducible representation of a Lie group of type $A$. In this paper, we provide a recursive formula for the number of reduced words of $w_0$ such that the corresponding string polytopes are combinatorially equivalent to a Gelfand-Cetlin polytope. The recursive formula involves the number of standard Young tableaux…
Over the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorithms for high-dimensional polyhedra given by inequalities. It is a famous and long standing open question in polyhedral theory and computational geometry whether the vertices of a polytope (bounded polyhedron), described by a set of linear constraints, can be enumerated in total polynomial time. In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra $P = \{x : A…
It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n by n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem. The unimodality of this sequance, which follows, was conjectured by Stanley (1983). Several generalizations are given.
We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers-Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behavior of the coefficients at $q = 1$ is Gaussian.
We call a real multi-dimensional array a {\em tensor} for short. In enumerating vertices of the polytopes of stochastic tensors, different approaches have been used: {(1)} Combinatorial method via Latin squares; {(2)} Analytic (topological) approach by using hyperplanes; {(3)} Computational geometry (polytope theory) approach; and (4) Optimization (linear programming) approach. As all these approaches are worthy of consideration and investigation in the enumeration problem, various bounds have been obtained. This note is to compare the existing upper bounds arose from different approaches.
$λ$-Chromatic Polynomials and Polytope Geometry
本文研究图的$λ$-色多项式，该多项式计数使用给定有限颜色集的不同$L(2,1)$-着色数量，通过内部-外部多面体框架下的格点枚举给出组合解释，并利用格路枚举和块-间隙技术计算完全图、完全二部图及多部图的$λ$-色多项式。
We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.
An Explicit Formula for Vertex Enumeration in the CUT(n) Polytope via Probabilistic Methods
本文通过概率视角和二进制编码递归构造，引入具有二的幂复合不变性的交替循环函数，给出了经典割多胞体 CUT(n) 顶点的显式闭式枚举公式。
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 …
Let $Q_n$ denote the $n$-dimensional hypercube with the vertex set $V_n=\{0,1}^n$. A 0/1-polytope of $Q_n$ is a convex hull of a subset of $V_n$. This paper is concerned with the enumeration of equivalence classes of full-dimensional 0/1-polytopes under the symmetries of the hypercube. With the aid of a computer program, Aichholzer completed the enumeration of equivalence classes of full-dimensional 0/1-polytopes for $Q_4$, $Q_5$, and those of $Q_6$ up to 12 vertices. In this paper, we present a method to compute the number of equivalence classes of full-dimensional 0/1-polytopes of $Q_n$ with…
We consider the semigroup S of highest weights appearing in tensor powers V^k of a finite dimensional representation V of a connected reductive group. We describe the cone generated by S as the cone over the weight polytope of V intersected with the positive Weyl chamber. From this we get a description for the asymptotic of the number of highest weights appearing in V^k in terms of the volume of this polytope.
In this article we disprove the conjectures asserting the positivity of the coefficients of the Ehrhart polynomial of matroid polytopes by De Loera, Haws and Köppe (2007) and of generalized permutohedra by Castillo and Liu (2015). We prove constructively that for every $n\geq 19$ there exist connected matroids on $n$ elements that are not Ehrhart positive. Also, we prove that for every $k\geq 3$ there exist connected matroids of rank $k$ that are not Ehrhart positive. Our proofs rely on our previous results on the geometric interpretation of the operation of circuit-hyperplane relaxation and o…
Counting Lattice Points in Minkowski Sums of Cross Polytopes
本文研究交叉多面体闵可夫斯基和的格点计数，引入支撑计数函数，利用根多面体对称性证明相关猜想，导出格点数量公式并推导Ehrhart多项式等相关结果。
Transfer Matrices and Ehrhart Theory for Path and Cyclic Block Polytopes
研究变量分块的块多面体，对路径排列的块在长度和扩张方向进行转移矩阵枚举与埃尔哈特理论分析，给出相关公式，还研究了循环类似物，阐述了路径和循环多面体的一些性质及相关组合解释与开放问题。
The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi…
Equivariant Ehrhart theory, commutative algebra and invariant triangulations of polytopes
本文发展等变Ehrhart理论，通过建立与交换代数的联系并探讨格多胞体存在不变格三角剖分的条件，推广了经典Ehrhart理论以处理有限群作用下的格点计数问题。
The $k$-tiling problem for a convex polytope $P$ is the problem of covering $\mathbb R^d$ with translates of $P$ using a discrete multiset $Λ$ of translation vectors, such that every point in $\mathbb R^d$ is covered exactly $k$ times, except possibly for the boundary of $P$ and its translates. A classical result in the study of tiling problems is a theorem of McMullen that a convex polytope $P$ that 1-tiles $\mathbb R^d$ with a discrete multiset $Λ$ can, in fact, 1-tile $\mathbb R^d$ with a lattice $\mathcal L$. A generalization of McMullen's theorem for $k$-tiling was conjectured by Gravin, …
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible point…
Stack-sorting simplices: geometry and lattice-point enumeration
本文研究由置换栈排序产生的多面体，重点分析 $Ln1$ 置换，证明其对应多面体为单纯形，并证明特定置换 $23\cdots n1$ 的栈排序凸包与单位立方体和 lecture-hall 单纯形具有相同格点计数函数，进而证明相关单纯形是指标为 2 的 Gorenstein 单纯形。
Formally Certifying the Vertex Set of a Polyhedron Faster than Informal Enumeration
本文提出基于Rocq的形式化验证多面体顶点集的方法，其速度比lrslib的非正式顶点枚举快1.5至5倍以上，可验证顶点列表的完整性或精确性。
A graph whose nodes have degree 1 or 3 is called a $\{1,3\}$-graph. Liu and Osserman associated a polytope to each $\{1,3\}$-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set $\{0,\frac14,\frac12,1\}$, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasi-polynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise. In the process of proving this theorem, several interes…
Underlying Flag Polymatroids
本研究通过关联贪心算法几何与单调路径多面体，建立拟阵与基础旗拟阵的几何关系，将构造推广至多拟阵，证明相关多面体为简单多面体，还探讨了其在环面簇中的应用及组合结构与单调路径的杨表计数关联。
A Closer Look at Chapoton's q-Ehrhart Polynomials
本文通过Brion定理的视角审视Chapoton关于q-Ehrhart多项式的研究，得出了包括显式公式、首项系数及渐近行为在内的多项精细化与扩展结果，并证明了有理多面体的类似结构定理与互反定理。
Boundary $h^\ast$-vectors and unimodular triangulations
本文通过正则单模三角剖分和 toric 理想的 Gröbner 衰减研究了格点多面体边界上的 $h^\ast$-多项式，推导了边界 $h^\ast$-多项式与正则单模三角剖分 $h$-多项式的关系，并建立了德恩-索默维尔型关系。
Transfer-Matrix Methods originated in physics where they were used to count the number of allowed particle states on a structure whose width $n$ is a parameter. Typically, the number of states is exponential in $n.$ One more mathematical instance of this methodology is to enumerate the proper vertex colorings of a graph of growing size by a fixed number of colors. In Ehrhart theory, lattice points in the dilation of a fixed polytope by a factor $k$ are enumerated. By inclusion-exclusion, relevant conditions on how the lattice points interact with hyperplanes are enforced. Typically, the number…
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