2. Ehrhart $h^\ast$polynomial: Unimodality and RealRootedness

Problem 2.02.
Does the PitmanStanley polytope have realrooted $h^*$polynomial?
Remark. [Alejandro Morales] In unpublished work of (BenedettiGonzalez D’LeonHanusaHarris) the \(h^*\) polynomial of the PitmanStanley polytope of dimension \(n\) is $h^*(z)=\sum_{\pi \in PF_n} z^{des(\pi)}$, where $PF_n$ is the set of parking functions of size $n$ and $des(\pi)$ is the number of descents of $\pi$ which are the indices $i$ such that $\pi(i)\geq \pi(i+1)$.


Problem 2.06.
For a lattice polytope $P$, let $P^n$ denote the Cartesian product of $P$ with itself $n$ times ($n\in \N$). Investigate what happens to the $h^*$polynomial of $P^n$. Does it eventually become realrooted? 
Problem 2.08.
For a symmetric edge polytope, can we obtain regular unimodular triangulations using liftings with low heights? If so, how many such triangulations can be obtained? 
Problem 2.1.
What are the ways of “measuring” root polytopes, i.e., edge polytopes?
Furthermore, how should Grothendieck polynomials be specialized to obtain Ehrhart $h^*$polynomials? 
Problem 2.12.
Consider the order polynomial, denoted $\Omega(P,t)$, of a finite poset $P$. Let $n$ be the number of elements in $P$.
Conjecture [Kahn, Saks] $\frac{\Omega(P,t)}{t^n}$ is nonincreasing for all $t\in \Z_{>0}$.
This statement is known to be true if all coefficients of $\Omega(P,t)$ are positive. Investigate the conjecture more generally.
(See solution to Exercise 3.163(b) inEnumerative Combinatorics Vol. 1
by Stanley). 
Definition: Let $P$ be a lattice polytope of dimension $d$. Let $\omega\in \Q[x_1,\ldots,x_d]$ be a homogeneous polynomial. We define the $\omega$Ehrhart function $$i(P;\omega;n) := \sum_{\alpha\in nP\cap \Z^d} \omega(\alpha).$$
Problem 2.14.
Note that $i(P;1;n) = i(P;n)$. It is known that $i(P;\omega;n)$ is a polynomial with leading coefficient given by $\int_P \omega$. Investigate $\omega$Ehrhart function for wellknown classes of polytopes. 
Problem 2.16.
Identify polytopes with extremal $h^*$vectors. That is, for a family of polytopes $\mathcal{P}$ and index $j$, find $P\in \mathcal{P}$ that minimize or maximize $h_j^*$. Potential families of interest include symmetric edge polytopes for a fixed degree sequence and simplices in Hermite normal form with fixed diagonal. 
Problem 2.18.
What relationship, if any, is there between heights of Hilbert basis elements and differences in $h^*$coefficients? For example, consider $h^*_{j+1}  h^*_j$ or more generally $h^*_j  h^*_i$. 
Problem 2.2.
Take the nonnegative hull of $h^*$vectors of polytopes in a fixed dimension. Does this form a polyhedral cone? 
Problem 2.22.
Are there “holes” in the set of $h^*$vectors of polytopes in a fixed dimension which are not a result of failing known linear inequalities? 
Problem 2.26.
Consider the convex hull of parking functions. Does this admit a unimodular triangulation? Furthermore, can the Ehrhart polynomial be obtained from this? 
Problem 2.28.
 (a) Find $0/1$polytopes with nonunimodal, nonlogconcave, etc. $h^*$vectors.
 (b) Find $0/1$polytopes with nonunimodal, nonlogconcave, etc. $f$vectors.
 (c) Find $0/\pm1$polytopes with nonunimodal, nonlogconcave, etc. $h^*$vectors.

Problem 2.3.
Determine a Veronese construction for rational polytopes.
Remark. [Mariel Supina] I want to clarify what I meant with this problem. For lattice polytopes, we can apply a certain transformation to the $h^*$polynomial of $P$ to get the $h^*$polynomial of $kP$ for a positive integer $k$ (see e.g. https://arxiv.org/pdf/0712.2645.pdf). Does such a transformation exist for rational polytopes? Does such a transformation exist in the setting of equivariant Ehrhart theory?


Problem 2.32.
How many chambers are there for transportation polytopes for $n\times n$ matrices? For example, how many Ehrhart polynomials are needed? 
Problem 2.38.
Investigate Stapledon’s Effectiveness Conjecture.
Conjecture. [Stapledon] Let $P$ be a lattice polytope invariant under the action of a group $G$. Then the characters of the equivariant $H^*$series is effective if and only if the equivariant $H^*$series is a polynomial.
Stapledon proved that $H^*$ effective implies $H^*$ is a polynomial. The converse remains open. 
Problem 2.4.
A spinal graph $G$ is a connected digraph on vertex set $V=\{1,2,...,n\}$ with edge multiset $E$, where all edges are directed from smaller to larger vertices and $E$ contains all edges of the form $(i,i+1)$ for $i=1,\ldots,n1$ (i.e., the “spine” of $G$).
What can be said about the $h^*$polynomials of flow polytopes of spinal graphs?
Conjecture: descent polynomial of lower $G$permutations $\leq h^*(\mathcal{F}_1;z) \leq $ descent polynomial of upper $G$permutations, where $\leq$ indicates domination.
Cite this as: AimPL: Ehrhart polynomials: inequalities and extremal constructions, available at http://aimpl.org/ehrhartineq.