Sam Payne
We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision and "local contributions" of the links of its nonunimodular faces. We also compute new examples of nonunimodal h^*-vectors of reflexive polytopes.
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Ehrhart non-positivity and unimodular triangulations for classes of s-lecture hall simplices
本文研究了s-讲座厅单纯形的Ehrhart非正性及unimodular三角剖分，提出了新的序列类并扩展了已知的三角剖分类。
Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices
该研究针对两类Hermite标准形单形，构造正则单模三角剖分，推导$h^\ast$-多项式等闭式公式，证明Ehrhart正性，还得到单行情形Ehrhart多项式非单峰的维度条件。
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…
Ehrhart Theory of the Join of Two Lattice Polytopes
受两个格点多面体笛卡尔积研究的启发，本文研究两个格点多面体联合的Ehrhart理论，解决了美国数学研究所网站上的一个著名开放问题。
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。
Sylvester simplices: Triangulations and Ehrhart-theoretic aspects
本文研究 Sylvester 单纯形的三角剖分与 Ehrhart 理论性质，证明其 h* 向量单峰，确定 f* 向量部分值，并验证其在低维的 Ehrhart 魔正性。
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 …
We use the ordinary Euler operator to compute the Ehrhart series for an arbitrary lattice polytope. The resulting formula involves the coefficients of the Ehrhart polynomial, combined via Eulerian numbers. We use this to compute $h^*_{d-1}$ in terms of the coefficients of the Ehrhart polynomial, resulting in a new linear inequality satisfied by the coefficents of the Ehrhart polynomial.
Ehrhart Properties under Operations on Lattice Polytopes
本文研究格多胞体上八种运算（如笛卡尔积、闵可夫斯基和等）对三类Ehrhart性质（系数正性、$h^*$-多项式系数性质、几何性质）的保持性，通过建立保持定理或构造反例，并探讨非保持情形下的充分或等价条件。
For any lattice polytope $P$, we consider an associated polynomial $\barδ_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart $δ$-vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice poly…
Series-parallel extensions and Ehrhart positivity
本文证明串并联拟阵及其多路径扩展的 Ehrhart 正性，推导系数公式与格点计数递归，并构造反例否证相关猜想。
Several properties of summatory Ehrhart polynomials and series of convex lattice polytopes
本文研究凸格点多面体的求和型 Ehrhart 多项式与级数，证明其基本性质，包括 Ehrhart-Macdonald 互反律的求和型类比。
The Ehrhart ring of the edge polytope $\mathcal{P}_G$ for a connected simple graph $G$ is known to coincide with the edge ring of the same graph if $G$ satisfies the odd cycle condition. This paper gives for a graph which does not satisfy the condition, a generating set of the defining ideal of the Ehrhart ring of the edge polytope, described by combinatorial information of the graph. From this result, two factoring properties of the Ehrhart series are obtained; the first one factors out bipartite biconnected components, and the second one factors out a even cycle which shares only one edge wi…
The Ehrhart series of alcoved polytopes
本文提出通过 alcove 的特定壳化顺序计算任意 alcoved 多胞体的 Ehrhart 级数的方法，并揭示了 Early 的装饰有序集合划分与超单纯形 $\Delta_{2,n}$ 壳化顺序之间的关联。
An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal $h^*$-vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider integrally closed reflexive simplices and discuss an operation that preserves reflexivity, integral closure, and unimodality of the $h^*$-vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements.
We introduce the definition of an Ehrhart limit, that is, a formal power series with integer coefficients that is the limit in the ring of formal power series of a sequence of Ehrhart $h^*$-polynomials. We identify a variety of examples of sequences of polytopes that yield Ehrhart limits, with a focus on reflexive polytopes and simplices.
In the present paper, we introduce a multibasic extension of the Ehrhart theory. We give a multibasic extension of Ehrhart polynomials and Ehrhart series. We also show that an analogue of Ehrhart reciprocity holds for multibasic Ehrhart polynomials.
The fractional stable set polytope ${\rm FRAC}(G)$ of a simple graph $G$ with $d$ vertices is a rational polytope that is the set of nonnegative vectors $(x_1,\ldots,x_d)$ satisfying $x_i+x_j\le 1$ for every edge $(i,j)$ of $G$. In this paper we show that (i) The $δ$-vector of a lattice polytope $2 {\rm FRAC}(G)$ is alternatingly increasing; (ii) The Ehrhart ring of ${\rm FRAC}(G)$ is Gorenstein; (iii) The coefficients of the numerator of the Ehrhart series of ${\rm FRAC}(G)$ are symmetric, unimodal and computed by the $δ$-vector of $2 {\rm FRAC}(G)$.
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.
It is known that the normalized volume of standard hypersimplices (defined as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give another generalization to colored permutations, using the flag statistics introduced by Foata and Han. We obtain in particular a new proof of Stanley's conjecture, and some combinatorial identities relati…
For every positive integer $n$, consider the linear operator $\U_{n}$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\frac{h(t)}{(1 - t)^{d + 1}} = \sum_{m \geq 0} g(m) t^{m}$, for some polynomial $g(m)$ with rational coefficients, then $\frac{\U_{n}h(t)}{(1- t)^{d + 1}} = \sum_{m \geq 0} g(nm) t^{m}$. We show that there exists a positive integer $n_{d}$, depending only on $d$, such that if $h(t)$ is a polynomial of degree at most $d$ with nonnegative integer coefficients and $h(0) \geq 1$, then for $n \geq n_{d}$, $\U_{n}h(t)$ has simple, real,…
We investigate the arithmetic-geometric structure of the lecture hall cone \[ L_n \ := \ \left\{λ\in \mathbb{R}^n: \, 0\leq \frac{λ_1}{1}\leq \frac{λ_2}{2}\leq \frac{λ_3}{3}\leq \cdots \leq \frac{λ_n}{n}\right\} . \] We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial is given by the $(n-1)$st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $L_n$, we conclude with observations and a conjecture regarding the structure of…
The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix cases, we give Pick-type formulas in terms of triangulations of a lattice polygon. As our main tool, we introduce $h^r$-tensor polynomials, extending the notion of the Ehrhart $h^\ast$-polynomial, and, for matrices, investigate their coefficients for positive semidefiniteness. In contrast to the usual $h^\ast$-polynomial, the coefficients are in general not monot…
We describe several experimental results obtained in four candidates social choice elections. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes.
We use machine learning to predict the dimension of a lattice polytope directly from its Ehrhart series. This is highly effective, achieving almost 100% accuracy. We also use machine learning to recover the volume of a lattice polytope from its Ehrhart series, and to recover the dimension, volume, and quasi-period of a rational polytope from its Ehrhart series. In each case we achieve very high accuracy, and we propose mathematical explanations for why this should be so.
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。
We present lower bounds for the coefficients of Ehrhart polynomials of convex lattice polytopes in terms of their volume. Concerning the coefficients of the Ehrhart series of a lattice polytope we show that Hibi's lower bound is not true for lattice polytopes without interior lattice points. The counterexample is based on a formula of the Ehrhart series of the join of two lattice polytope. We also present a formula for calculating the Ehrhart series of integral dilates of a polytope.
Harmonics and graded Ehrhart theory
本文基于调和空间与 Macaulay 逆系统引入了 Ehrhart 级数的 $q$-形变，猜想其为有理函数，并构建了 Hilbert 级数与之匹配的双分次代数，同时证明了关于点配置 Minkowski 和的逆系统新结果。
Weighted Ehrhart series and a type-$\mathsf{B}$ analogue of a formula of MacMahon
本文通过多面体的$h^*$-多项式给出了带符号多重集排列上主指标与下降统计量联合分布的生成多项式公式，并建立了A、B型广义欧拉数与多面体之间的联系，进而将结果推广到着色排列。
We describe an algorithm for the computation of generalized (or weighted) Ehrhart series based on Stanley decompositions as implemented in the offspring NmzIntegrate of Normaliz. The algorithmic approach includes elementary proofs of the basic results. we illustrate the computations by examples from combinatorial voting theory.
The Ehrhart series of magic squares of orders seven and eight
研究七阶幻方的埃尔哈特级数，通过SimpCone分解等方法确定其为既约有理函数，利用多种技术计算相关多面体在有限域上的生成函数等，最终确定了级数的分母、分子等特性并证明相关结论。
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。
We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension d, a cross-polytope of dimension d, respectively a hypercube of dimension d<4, using commutative algebra. We deduce summation formulae for products of q-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.
Ehrhart Functions of Weighted Lattice Points
本文研究了凸多面体格点的不同加权方法，探讨了带权Ehrhart函数及其生成函数和相关环的代数与组合性质，提出q-加权、r-加权和s-加权Ehrhart函数，并证明了其一般性质及Ehrhart环的结构。
The type A_n full root polytope is the convex hull in R^{n+1} of the origin and the points e_i-e_j for 1<= i<j <= n+1. Given a tree T on the vertex set [n+1], the associated root polytope P(T) is the intersection of the full root polytope with the cone generated by the vectors e_i-e_j, where (i, j) is an edge of T, i<j. The reduced forms of a certain monomial m[T] in commuting variables x_{ij} under the reduction x_{ij}x_{jk} --> x_{ik}x_{ij}+x_{jk}x_{ik}+βx_{ik}, can be interpreted as triangulations of P(T). Using these triangulations, the volume and Ehrhart polynomial of P(T) are obtained. I…
We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a "long" regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenste…
Hibi showed that the polynomial in the numerator of the Ehrhart series of a reflexive polytope is palindromic. We proved that those in the numerator of the Ehrhart series of every graph polytope (defined later) of the bipartite graph is palindromic. From this, one of the conjectures (raised in the A205497 of OEIS \cite{[O]}) follows immediately.
Exact Ehrhart Series of Birkhoff Polytopes via Constant Terms and Finite-Field Evaluation
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。
A result of Haglund implies that the $(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector $(-n, 1, \dots, 1)$. We study the $(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at $t=1$, $0$, and $q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades and Sagan about the $(q, q^{-1})$-Ehrhart fun…
Alternating adjacent-sum polytopes: transfer matrices and Ehrhart series
研究一类周期为二的相邻和格多面体，通过转移矩阵等方法分析其格点数、生成函数等性质，包括奇偶分裂、递归关系等，还推导相关函数及恒等式，探讨了戈伦斯坦性质在不同\(s\)值下的情况。
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…
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。
Equivariant Hilbert and Ehrhart series under translative group actions
本文研究有限群在单纯复形和格多胞体上的表示，利用平移群作用与适当着色理论，证明等变 Hilbert 与 Ehrhart 级数的有理表达式，并应用于 alcoved 多胞体等实例。
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。
Unimodular Valuations beyond Ehrhart
本文提出了一种基于扩展无界多面体和基本不变量理论的方法，对整数点多边形的单模估值进行了完整分类。
Two families of Hermite normal form simplices
研究两类 Hermite 标准形单纯形的 Ehrhart 系数与整数分解性质，建立了单峰性并分类了对数凹情形，通过负连分数条件刻画了整数分解与三角剖分，解决了三个开放问题。
Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem
本文推广 Stanley 定理至加权 Ehrhart 级数，证明非负性与单调性在齐次多项式权重下成立，并针对线性形式平方给出二维强结果及反例。
Ehrhart polynomials, Hecke series, and affine buildings
本文研究格多面体Ehrhart多项式系数与辛Hecke代数的关系，证明相关函数是Hecke特征函数，并计算其在与p-adic辛群仿射建筑顶点上的局部zeta函数，证实Vankov猜想。
Recently several authors have proved results on Ehrhart series of free sums of rational polytopes. In this note we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from structural results on affine monoids and their algebras that allow conclusions for Hilbert and Ehrhart series. We characterize when a binomial regular sequence generates a prime ideal or even normality is preserved for the residue class ring.
With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable surfaces for every $q=3k$, $k\geq 2$, and every $q=3k+1$, $k\geq 3$. Series of cyclic tessellations of surfaces are derived from these triangulated series.
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…
Boundary $h^\ast$-polynomials of rational polytopes
本文提出通过多胞体边界的 $h^*$-多项式理解 Ehrhart 理论的新方法，统一恢复了现有必要条件并扩展至有理多胞体，且应用于 Gorenstein 多胞体与有理 Ehrhart 膨胀。
A geometric and generating function approach to plethysm
本文研究了plethysm系数的双变量生成函数，证明其有理性，并利用q-Ehrhart理论给出几何算法，同时展示了生成函数与量子Ehrhart系列的关系及SL₂-plethysm系数的线性递推关系。
Proofs of four generating function conjectures for arbor polytopes
本文证明了四个关于poset和特定arbor序列关联聚类体不变量的生成函数猜想，包括Zeta多项式、M-三角形、Ehrhart多项式和体积函数拉普拉斯变换的闭式公式。
We present two versions of a method for generating all triangulations of any punctured surface in each of these two families: (1) triangulations with inner vertices of degree at least 4 and boundary vertices of degree at least 3 and (2) triangulations with all vertices of degree at least 4. The method is based on a series of reversible operations, termed reductions, which lead to a minimal set of triangulations in each family. Throughout the process the triangulations remain within the corresponding family. Moreover, for the family (1) these operations reduce to the well-known edge contraction…
The 3-state Potts model on planar triangulations: explicit algebraic solution
本文通过显式代数方法确定了平面三角划分的3态Potts生成函数T(ν,w)的精确表达式，并证明了其临界值和指数特性，同时解决了关于立方图的猜想。
The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plucker relations from geometry. We give a solution to the Plucker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generat…
Inequalities for $f^*$-vectors of Lattice Polytopes
本文研究了格多面体 $f^*$-向量的系数不等式，揭示了其与单纯多面体 $f$-向量不等式的相似性，并证明了特定多面体族的单峰性及 $h^*$-向量相同条件下的单峰 $f^*$-向量存在性。
An algebra over the operad of posets and structural binomial identities
本文研究了序列平行 poset 的顺序序列生成函数，并基于其基的性质，推导了 Stanley 互易定理的新证明，同时发现新的二项式恒等式和负超几何分布的性质。
For an $n$-dimensional lattice simplex $Δ_{(1,\mathbf{q})}$ with vertices given by the standard basis vectors and $-\mathbf{q}$ where $\mathbf{q}$ has positive entries, we investigate when the Ehrhart $h^*$-polynomial for $Δ_{(1,\mathbf{q})}$ factors as a product of geometric series in powers of $z$. Our motivation is a theorem of Rodriguez-Villegas implying that when the $h^*$-polynomial of a lattice polytope $P$ has all roots on the unit circle, then the Ehrhart polynomial of $P$ has positive coefficients. We focus on those $Δ_{(1,\mathbf{q})}$ for which $\mathbf{q}$ has only two or three di…
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