Ehrhart Positivity in Lattice Polytopes

• Ehrhart positivity is the property that all coefficients in a lattice polytope’s Ehrhart polynomial are positive, reflecting its combinatorial and geometric structure.
• Families such as cubes, simplices, flow polytopes, and zonotopes exhibit this property through positive linear factorizations and explicit combinatorial summation formulas.
• Counterexamples like Reeve tetrahedra and certain order polytopes illustrate conditions where Ehrhart coefficients become negative, guiding deeper structural investigations.

Ehrhart positivity is a property of lattice polytopes characterized by the nonnegative nature of all coefficients in their Ehrhart polynomial, which enumerates lattice points in integral dilations of the polytope. Introduced in the context of enumerative combinatorics and convex geometry, Ehrhart positivity is deeply connected to diverse phenomena in algebraic, geometric, and combinatorial structures. Despite its appealing implications—such as connections to unimodality and real-rootedness in $h^*$-polynomials—Ehrhart positivity is not universal: while some remarkable families of polytopes always enjoy this property, explicit constructions reveal broad classes where positivity fails. The paper of Ehrhart positivity therefore centers on rigorous structural, enumerative, and algebraic criteria that guarantee positivity, as well as the identification and analysis of counterexamples and extremal sign patterns.

1. Definitions and Analytical Framework

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional integral (lattice) polytope. Ehrhart's theorem asserts that the counting function $i(P,t):=|tP \cap \mathbb{Z}^d|$ is a polynomial of degree $d$ in $t$: $$i(P, t) = c_d t^d + c_{d-1} t^{d-1} + \cdots + c_0.$$ The coefficients $c_d$ (proportional to the normalized volume of $P$) and $c_0$ (equal to $1$) are always positive; $c_{d-1}$ often—but not always—admits a geometric interpretation as half the total facet volume and is typically positive. However, the intermediate ("middle") coefficients can be negative, even for highly regular polytopes.

A polytope is Ehrhart positive if all coefficients $c_0, \ldots, c_d$ are positive. The central focus of this area is to understand which families of polytopes guarantee this condition, and by which mechanisms (combinatorial, algebraic, or geometric) positivity can or cannot arise.

Closely related is the $h^*$-vector (or $\delta$-vector), arising in the expansion of the Ehrhart series: $$\mathrm{Ehr}_P(z) = \sum_{t\geq0} i(P, t)z^t = \frac{h_0^* + h_1^*z + \ldots + h_d^*z^d}{(1-z)^{d+1}},$$ with all $h_j^*\geq 0$ by Stanley's nonnegativity theorem. There are explicitly known relations between the $h^*$-vector and the Ehrhart coefficients: $i(P, t) = \sum_{j=0}^{d} h_j^* \binom{t+d-j}{d}$. While $h^*$-vector entries are always nonnegative, their unimodality or real-rootedness is a focal point of paper, and—importantly—there is no general implication in higher dimensions between the positivity of the $h^*$-vector and Ehrhart positivity (Liu et al., 2018, Ferroni et al., 2023).

2. Families of Ehrhart Positive Polytopes

Distinct, nontrivially overlapping families of polytopes have been proven to be Ehrhart positive by a variety of arguments:

• Polytopes with Positive Linear Factorizations
  • Unit cube $\Box_d$: $i(\Box_d, t) = (t+1)^d$.
  • Standard simplex $\Delta_d$: $i(\Delta_d, t) = \binom{t+d}{d}$.
  • Each is a product of linear polynomials with positive coefficients, ensuring global positivity.
• Pitman–Stanley and Flow Polytopes
  • Pitman–Stanley polytopes and related flow polytopes have explicit "sum-of-products" formulas where each summand is a product of positive linear terms in $t$, implying Ehrhart positivity term-by-term (Liu, 2017). This encompasses cases handled by the Lidskii formula, Postnikov–Stanley decompositions, and Baldoni–Vergne residue methods.
• Zonotopes
  • Stanley's formula expresses Ehrhart coefficients of a zonotope as positive sums over linearly independent sets of line segments; combinatorial dissection into parallelepipeds provides further evidence (Ferroni, 2019).
• Root-Real-Parts Arguments
  • For families such as standard cross-polytopes and certain simplices (e.g., "reflexive simplices" and Payne's examples), if all roots of the Ehrhart polynomial have negative real part, all coefficients are positive. In several cases, this is established via transfer theorems from $h^*$-polynomials with roots on the complex unit circle (Liu, 2017).
• Fully Integral and Cyclic Polytopes
  • If a polytope is $k$-integral (all faces up to $k$-dimensional lie in integral affine spaces), it admits an explicit Ehrhart coefficient formula in terms of volumes/projections, guaranteeing positivity for cyclic and lattice-face polytopes.
• Generalized Permutohedra
  • Though not proven generally, all type-A generalized permutohedra with positive BV-$\alpha$ values (as in the Berline–Vergne valuation) are Ehrhart positive. A reduction theorem implies that positivity for the regular permutohedron extends to all in its class (Liu, 2017, Jochemko et al., 2019). For the linear coefficient, positivity holds for all integral generalized permutohedra (Jochemko et al., 2019).
• Catalan Matroids and Related Classes
  • $(a,b)$-Catalan matroid polytopes decompose as positive sums of uniform matroid polytopes, each Ehrhart positive by Ferroni's theorem (Ferroni, 2019, Chen et al., 25 Mar 2025).
• Shard Polytopes and Order Polytopes of Specific Posets
  • Shard polytopes of type A, and order polytopes of fence posets (including zig-zag and circular fence posets), have order polynomials with nonnegative coefficients, hence Ehrhart positivity (Ferroni et al., 20 Mar 2025).
• Rank 2 Matroid Polytopes
  • All Ehrhart polynomials for base polytopes of connected rank 2 matroids are positive and coefficient-wise bounded above and below by those of uniform and minimal matroids (Ferroni et al., 2021).

3. Negative Cases and Obstructions to Positivity

Despite many positive families, systematic constructions yield negative examples:

• Reeve Tetrahedra: The 3D Reeve tetrahedron $\mathcal{R}_m$, with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(1,1,m)$, has

$$i(\mathcal{R}_m, t) = \frac{m}{6} t^3 + t^2 + \frac{12-m}{6}t + 1.$$

The linear coefficient is negative for $m \geq 13$ (Liu, 2017, Ferroni et al., 2023).

• Order Polytopes: For the family $Q_k$ (poset with one minimal element covered by $k$ elements), the Ehrhart coefficients are given explicitly in terms of Bernoulli numbers, and the sign pattern can be computed: for $d \geq 21$, negative coefficients occur, with capacity for controlling the number and position of negative coefficients (Liu et al., 2018).
• High-Dimensional and Minkowski Sums: Smooth polytopes in dimension $\geq 7$ (via chiseling constructions) can have negative coefficients. Minkowski sums are not preservation operators for positivity: even when $P$ and $Q$ are Ehrhart positive, $P+Q$ may not be (Liu, 2017).
• Generalized Permutohedra and Matroids: Recent work disproved the conjecture that all matroid base polytopes (or all generalized permutohedra) are Ehrhart positive by explicitly constructing matroids (notably sparse paving matroids) in every rank $\geq 3$ (with $n\geq 19$ for ground set size) exhibiting negative coefficients (Ferroni, 2021).

4. Structural Connections, Techniques, and Formulas

The methodology dividing positive from nonpositive cases includes:

Technique/Property	Example/Family	Key Outcome
Factorization into positive linear terms	Cubes, simplices	Immediate positivity
Explicit sum/product/combinatorics	Zonotopes, flow polytopes, Pitman–Stanley	Structural combinatorial summation pos.
Real-rootedness/complex root analysis	Cross-polytopes, reflexive simplices	Roots $\Rightarrow$ coefficient positivity
BV-$\alpha$-valuation, McMullen’s Formula	Generalized permutohedra, Tesler polytopes	Local/global positivity if $\alpha$-values positive
Decomposition via inclusion–exclusion on faces/ideals	Catalan matroids, shard polytopes	Aggregation reduces to known positive subcases
Use of Bernoulli numbers (order polytopes)	Family $Q_k$	Explicit sign pattern control; negative middle coefficients

Key formulas:

• For the $d$-cube: $i(\Box_d, t) = (t+1)^d$
• For the simplex: $i(\Delta_d, t) = \binom{t+d}{d}$
• For flow polytopes: $$i(P,t) = \sum_j \prod_k \binom{a_k t + out_k}{j_k} K(j-out)$$
• In BV-$\alpha$ valuation: $$[t^k]i(P,t) = \sum_{\dim F = k} \alpha(F,P) \mathrm{nvol}(F)$$, positivity of all $\alpha(F,P)$ $\Rightarrow$ Ehrhart positivity
• In Catalan matroids: $$i(C_{a,b}, t) = \sum_{X \in \mathcal{X}} c_X \, i(U_X, t)$$, with $c_X \geq 0$ (Chen et al., 25 Mar 2025)

5. Relationship with $h^*$-Polynomials and Further Positivity Notions

• All $h^*$-vector entries are nonnegative, but the $h^*$-polynomial may fail to be unimodal or real-rooted even for Ehrhart positive polytopes (Liu et al., 2018, Ferroni et al., 2023).
• Magic Positivity (Editor's term): If the Ehrhart polynomial, expanded in the basis $\{n^i(1+n)^{d-i}\}$, has all coefficients nonnegative, then (a) the Ehrhart polynomial is Ehrhart positive, and (b) the $h^*$-polynomial is real-rooted, hence unimodal and log-concave (Konoike, 25 Sep 2024, Konoike, 30 Apr 2025, Ferroni et al., 2023).
• Dilating a polytope $P$ by sufficiently large $k$ forces $E_{kP}(n)$ to become magic positive, but no universal such $k$ exists in dimensions $d\geq3$ (Konoike, 30 Apr 2025).

6. Examples, Counterexamples, and Classification

• Zonotopes: Always Ehrhart positive (Ferroni, 2019, Liu et al., 2018).
• Uniform Matroids (Hypersimplices): Ehrhart positive with explicit combinatorial formulas in terms of weighted Lah numbers and Eulerian numbers (Ferroni, 2019).
• Minimal Matroids: Ehrhart positive, $h^*$-real-rooted; their role is extremal for Ehrhart coefficient bounds (Ferroni, 2020).
• Catalan Matroids (and variations): Ehrhart positive via decomposable formula into uniform matroids (Chen et al., 25 Mar 2025, Fan et al., 2021).
• Panhandle Matroids: Ehrhart positive, confirmed via enumeration of ordered chain forests; this proof model is extensible to paving matroids (Deligeorgaki et al., 2023).
• Order Polytopes: Negative middle coefficients occur in high dimensions, controlled by parity and divisibility via explicit Bernoulli number formulae (Liu et al., 2018).
• Matroid Counterexamples: Existence for all ranks $\geq 3$ and $n\geq19$; basis polytope construction via circuit-hyperplane relaxation and coding theory (Ferroni, 2021).

7. Open Problems and Future Research Directions

Open questions and ongoing conjectures include:

• Complete characterization of Ehrhart positivity for generalized permutohedra, type-B permutohedra, matroid polytopes outside the uniform and minimal cases (Liu, 2017, Jochemko et al., 2019, Fan et al., 2021).
• Positivity for Birkhoff polytopes (doubly stochastic matrices) and Tesler polytopes remains open in general, though partial lower-degree positivity for Tesler polytopes has been established (Lee et al., 2019).
• Explicit sign pattern control: For which dimension $d$, and for which subset of middle coefficients, can prescribed "sign signatures" (some negative, some positive) be achieved? Complete answers only known in low dimensions (Liu, 2017).
• Minkowski summation: Structural understanding of when the Minkowski sum of Ehrhart positive polytopes is Ehrhart positive; counterexamples exist (Liu, 2017).
• Combinatorial interpretation of Ehrhart coefficients: Rich combinatorial models such as ordered chain forests are increasingly being leveraged to obtain structural and computational control over coefficients (Deligeorgaki et al., 2023).
• Geometric and representation-theoretic connections: For instance, links to stretched Littlewood–Richardson coefficients and hives are not yet fully understood (Liu, 2017).

8. Broader Implications and Structural Impact

The research on Ehrhart positivity reveals complex and subtle interactions between the geometry of polytopes, their combinatorial models, and algebraic properties of associated generating functions. Positive families often correspond to rich combinatorial structures (e.g., uniform matroids, Catalan and shard polytopes, Pitman–Stanley polytopes), while negative examples highlight the sensitivity to global and local combinatorial configuration, particularly as manifested in high dimensions or as a result of specific geometric operations (e.g., relaxation, chiseling, Minkowski summation). Importantly, results that combine geometric analysis (e.g., BV-$\alpha$-valuation, McMullen's formula), algebraic transformation (e.g., magic positivity expansion, real-rootedness analysis), and deep combinatorial enumeration (e.g., chain forests, Lah number formulas) drive the classification and understanding of Ehrhart positivity and reveal directions for the future paper of polyhedral and algebraic combinatorics.