The Ehrhart Function for Symbols

带权格点的Ehrhart函数

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。

Reconstruction of rational polytopes from the real-parameter Ehrhart function of its translates

When extending the Ehrhart lattice point enumerator $L_P(t)$ to allow real dilation parameters $t$, we lose the invariance under integer translations that exists when $t$ is restricted to be an integer. This paper studies this phenomenon; in particular, it is shown that, for full-dimensional $P$, not only there are infinitely many different functions $L_{P + w}(t)$ (for integer $w$), but that for rational $P$ the collection of these functions identifies $P$ uniquely.

An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in combinatorics, Ehrhart's method for proving that a couting function is a polynomial, the connection between polyhedral cones, rational functions and quasisymmetric functions, methods for bounding coefficients, combinatorial reciprocity theorems, algorithms for counting integer points in polyhedra and computing rational function representations, as well as visualizations …

分段指数函数与Ehrhart扇

Piecewise-exponential functions and Ehrhart fans

本文引入Ehrhart扇概念，证明完备幺模扇和拟阵的Bergman扇均为Ehrhart扇，且其Ehrhart泛函分别对应格点计数和拟阵Euler示性数，并证明该性质仅依赖于扇的支撑集。

Maximal Periods of (Ehrhart) Quasi-Polynomials

A \emph{quasi-polynomial} is a function defined of the form $q(k) = c_d(k) k^d + c_{d-1}(k) k^{d-1} + ... + c_0(k)$, where $c_0, c_1, ..., c_d$ are periodic functions in $k \in \Z$. Prominent examples of quasi-polynomials appear in Ehrhart's theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the $c_j(k)$ for Ehrhart quasi-polynomials. For generic polytopes, McMullen's bounds seem to be sharp, but sometimes smaller periods exist. We prove that the second leading coefficient of an Ehrhart quasi-polynomial always has maximal expec…

A bijective proof for a theorem of Ehrhart

We give a new proof for a theorem of Ehrhart regarding the quasi-polynomiality of the function that counts the number of integer points in the integral dilates of a rational polytope. The proof involves a geometric bijection, inclusion-exclusion, and recurrence relations, and we also prove Ehrhart reciprocity using these methods.

从驯顺函数的 Hodge 理论到多面体的 Ehrhart 理论

From Hodge theory for tame functions to Ehrhart theory for polytopes

本文通过分析 Sabbah 混合 Hodge 结构中 Hodge 滤过的 Poincaré 多项式性质，建立正则函数 Hodge 理论与多面体 Ehrhart 理论之间的联系。

Ehrhart-Macdonald reciprocity extended

For a convex polytope P with rational vertices, we count the number of integer points in integral dilates of P and its interior. The Ehrhart-Macdonald reciprocity law gives an intimate relation between these two counting functions. A similar counting function and reciprocity law exists for the sum of all solid angles at integer points in dilates of P. We derive a unifying generalization of these reciprocity theorems which follows in a natural way from Brion's Theorem on conic decompositions of polytopes.

正广面多胞形的Ehrhart $h^*$-多项式

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。

Coprime Ehrhart theory and counting free segments

A lattice polytope is "free" (or "empty") if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $α_i(P;n)$ that count the number of free polytopes in $nP$ with $i$ vertices. For $i=1$, this is the famous Ehrhart polynomial. For $i > 3$, the computation is likely impossible and for $i=2,3$ computationally challenging. In this paper, we develop a theory of coprime Ehrhart functions, that count lattice points with relatively prime coordinates, and use it to compute $α_2(P;n)$ for unimodular simplices. We show that…

通过埃赫拉特理论证明Watanabe-Yoshida的猜想

A Proof of a conjecture of Watanabe--Yoshida via Ehrhart Theory

本文通过埃赫拉特理论提供了Watanabe-Yoshida猜想的另一种证明，利用新的显式组合公式和生成函数的Hadamard积。

New irrational polygons with Ehrhart-theoretic period collapse

In a recent paper, Cristofaro-Gardiner--Li--Stanley [CGLS15] constructed examples of irrational triangles whose Ehrhart functions (i.e. lattice-point count) are polynomials when restricted to positive integer dilation factors. This is very surprising because the Ehrhart functions of rational polygons are usually only quasi-polynomials. We demonstrate that most of their triangles can also be obtained by a simple cut-and-paste procedure that allows us to build new examples with more sides. Our examples might potentially have applications in the theory of symplectic embeddings.

Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the res…

有理多面体Ehrhart系数的周期性

Periods of Ehrhart coefficients of rational polytopes

研究有理多面体Ehrhart系数的周期性，通过构造多面体族来探索准多项式的周期特性。

Volumes and Ehrhart polynomials of flow polytopes

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…

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.

来自 Ehrhart 多项式的辛 Hecke 特征基

Symplectic Hecke eigenbases from Ehrhart polynomials

本文研究提取格多面体Ehrhart多项式系数的函数，证明其在偶数维度下为辛Hecke特征函数，并应用球函数及zeta函数理论推导出相关算术函数的解析、渐近与组合结果。

Counting integer points of flow polytopes

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…

Asymptotic Distribution Of The Roots Of The Ehrhart Polynomial Of The Cross-Polytope

We use the method of steepest descents to study the root distribution of the Ehrhart polynomial of the $d$-dimensional cross-polytope, namely $\mathcal{L}_{d}$, as $d\rightarrow \infty$. We prove that the distribution function of the roots, approximately, as $d$ grows, by variation of argument of the generating function $\sum_{m\geq 0}\mathcal{L}_{d}(m)t^{m+x-1}=(1+t)^{d}(1-t)^{-d-1}t^{x-1}$, as $t$ varies appropriately on the segment of the imaginary line contained inside the unit disk.

泛型正交体的Ehrhart多项式

Ehrhart Polynomials of Generic Orthotopes

本文为整泛型正交体建立Ehrhart多项式理论，揭示其格点数与各类花型单位立方体数量之间的关系，并通过为只读一次布尔函数引入局部多项式来实现。

Residues formulae for volumes and Ehrhart polynomials of convex polytopes

In these notes, we explain residue formulae for volumes of convex polytopes, and for Ehrahrt polynomials based on the notion of total residue. We apply this method to the computation of the volume of the Chan-Robbins polytope. The final computation is based on a total residue formula for the system $A_n$, similar to Morris identity. For flow polytopes, a formula of change of variables in total residues leads to a "nice formula" for Ehrhart polynomials in function of mixed volumes. We apply it to Pitman-Stanley polytope.

相对埃哈特理论I：相对埃哈特最终多项式

Relative Ehrhart functions: eventual polynomiality and dualities

该研究扩展经典埃哈特理论，用几何对象Q替代格点，证明相关计数函数具有最终拟多项式性，首项为vol(P)t^d，结果由维度归纳推导得出。

通过Ehrhart理论研究二次曲面的Hilbert-Kunz乘数

Hilbert-Kunz multiplicity of quadrics via Ehrhart theory

本文通过Ehrhart多项式研究二次曲面的Hilbert-Kunz乘数，证明其为特征p的有理函数，并揭示其随维度变化的性质。

Ehrhart series and lattice triangulations

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.

Ehrhart positivity of Tesler polytopes and Berline-Vergne's valuation

For $\ba \in \R_{\geq 0}^{n}$, the Tesler polytope $\tes_{n}(\ba)$ is the set of upper triangular matrices with non-negative entries whose hook sum vector is $\ba$. Motivated by a conjecture of Morales', we study the questions of whether the coefficients of the Ehrhart polynomial of $\tes_n(1,1,\dots,1)$ are positive. We attack this problem by studying a certain function constructed by Berline-Vergne and its values on faces of a unimodularly equivalent copy of $\tes_n(1,1,\dots,1).$ We develop a method of obtaining the dot products appeared in formulas for computing Berline-Vergne's function d…

有理多面体通过实数扩张的Ehrhart准多项式

Ehrhart quasi-polynomials of rational polytopes by real dilations

本文研究有理n维多面体P的Ehrhart函数L(P,t)，证明其为实变量t的准多项式，并给出有理单纯形的系数函数显式表达式，同时保持对偶律。

Ehrhart多项式、Hecke级数与仿射建筑

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。

Flow polytopes and the space of diagonal harmonics

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…

马尔可夫三角形的周期崩溃

Period collapse of Markov triangles

本文通过积分仿射几何方法，将斐波那契三角形的周期崩溃结果推广到所有马尔可夫三角形，并展示了强周期崩溃的新实例。

Weighted lattice point sums in lattice polytopes, unifying Dehn--Sommerville and Ehrhart--Macdonald

Let $V$ be a real vector space of dimension $n$ and let $M\subset V$ be a lattice. Let $P\subset V$ be an $n$-dimensional polytope with vertices in $M$, and let $φ\colon V\rightarrow \CC $ be a homogeneous polynomial function of degree $d$ (i.e., an element of $\Sym^{d} (V^{*})$). For $q\in \ZZ_{>0}$ and any face $F$ of $P$, let $D_{φ,F} (q)$ be the sum of $φ$ over the lattice points in the dilate $qF$. We define a generating function $G_φ(q,y) \in \QQ [q] [y]$ packaging together the various $D_{φ,F} (q)$, and show that it satisfies a functional equation that simultaneously generalizes Ehrhart…

幸运与魔法：Pitman-Stanley多面体与停车函数

Luck and magic for Pitman-Stanley polytopes and parking functions

本文研究Pitman-Stanley多面体的Ehrhart理论，证明其Ehrhart多项式具有强正性现象，进而推导出其h*-多项式实根性。通过解读系数于magic基中，关联到修改停车协议中的幸运汽车数。

关于plethysm的几何和生成函数方法

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。

有理多胞形的 Ehrhart $h^*$-多项式的分解

Decompositions of Ehrhart $h^*$-polynomials for rational polytopes

本文为有理多胞形的 Ehrhart $h^*$-多项式提供了两种分解公式，推广了 Betke-McMullen 定理与 Stapledon 结果，并据此给出了 Stanley 单调性定理的新证明及 Stanley 和 Hibi 不等式的有理扩展。

宇宙学多面体的 Ehrhart 理论

Ehrhart theory of cosmological polytopes

本文研究宇宙学多面体的 Ehrhart $h^\ast$-多项式，给出图不交并与 $1$-和的递归公式、次数、回文性及紧下界，并推广归一化体积结果。

Three Ehrhart Quasi-polynomials

Let $P(b)\subset R^d$ be a semi-rational parametric polytope, where $b=(b_j)\in R^N$ is a real multi-parameter. We study intermediate sums of polynomial functions $h(x)$ on $P(b)$, $$ S^L (P(b),h)=\sum_{y}\int_{P(b)\cap (y+L)} h(x) \mathrm dx, $$ where we integrate over the intersections of $P(b)$ with the subspaces parallel to a fixed rational subspace $L$ through all lattice points, and sum the integrals. The purely discrete sum is of course a particular case ($L=0$), so $S^0(P(b), 1)$ counts the integer points in the parametric polytopes. The chambers are the open conical subsets of $R^N$ s…

交替相邻和多面体：转移矩阵与埃尔哈特级数

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。

Intermediate Sums on Polyhedra II: Bidegree and Poisson Formula

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where s is an arbitrary real vertex and c is a rational polyhedral cone. For a given rational subspace L, we integrate a given polynomial function h over all lattice slices of the affine cone s + c parallel to the subspace L and sum up the integrals. We study these …

格切片、埃尔哈特多项式与广义停车函数多面体的神奇正性

Lattice slices, Ehrhart polynomials, and magic positivity of generalized parking-function polytopes

研究广义停车函数多面体，证明其格切片性质并得出格点数递归，推导埃尔哈特多项式公式，在特殊情况有显式形式与生成函数，还对双参数族多面体的神奇正性分类，回答相关问题并扩展前人工作。

New examples of period collapse

"Period collapse" refers to any situation where the period of the Ehrhart function of a polytope is less than the denominator of that polytope. We study several interesting situations where this occurs, primarily involving triangles. For example: 1) we determine exactly when the Ehrhart function of a right triangle with legs on the axes and slant edge with irrational slope is a polynomial; 2) we find triangles with periods given by any even-index k-Fibonacci number, and larger denominators; 3) we construct several higher dimensional examples. Several related issues are also discussed, includin…

A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope

We present a multivariate generating function for all n x n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B_n of n x n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B_n and any of its faces.

Lattice polytopes having h^*-polynomials with given degree and linear coefficient

The h^*-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h^*-polynomial of degree d and with linear coefficient h^*_1. We show that P has to be a lattice pyramid over a lower-dimensional lattice polytope, if the dimension of P is greater or equal to h^*_1 (2d+1) + 4d-1. This result has a purely combinatorial proof and generalizes a recent theorem of Batyrev.

关于计数函数（denumerant functions）的三个简单约化公式

Three Simple Reduction Formulas for the Denumerant Functions

本文利用 Bernoulli-Barnes 多项式，将 Ehrhart 等人针对三参数且元素两两互素情形建立的受限分拆函数约化公式推广到一般集合 $A$。

Discrete sums of classical symbols on Z^d and zeta functions associated with Laplacians on tori

We prove the uniqueness of a translation invariant extension to non integer order classical symbols of the ordinary discrete sum on $L^1$-symbols, which we then describe using an Hadamard finite part procedure for sums over integer points of infinite unions of nested convex polytopes in $\R^d$. This canonical regularised sum is the building block to construct meromorphic extensions of the ordinary sum on holomorphic symbols. Explicit formulae for the complex residues at their poles are given in terms of noncommutative residues of classical symbols, thus extending results of Guillemin, Sternber…

Automorphic properties of generating functions for generalized rank moments and Durfee symbols

We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta func…