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1. Ehrhart Positivity


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      Conjecture 1.05.

      Let $[n]:=\{1,\ldots,n\}$ be a groundset, and let $i(P; t)$ denote the Ehrhart polynomial of a lattice polytope $P$. Then coefficient-wise, $$i(T_{r,n}; t) \leq i(\mathsf{Pan}_{s,r,n};t) \leq i(U_{r,n};t),$$ where $T_{r,n}$ is the rank $r$ minimal matroid on $[n]$, $U_{r,n}$ is the rank $r$ uniform matroid on $[n]$, and $\mathsf{Pan}_{s,r,n}$ is the rank $r$ Panhandle matroid on $[n]$.


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          Problem 1.1.

          What is known (or can be shown) about the Ehrhart positivity of polytopes related to Birkhoff polytopes (e.g., transportation polytopes)?


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              Problem 1.15.

              Are $d$-dimensional smooth polytopes with $\leq d+3$ facets Ehrhart positive or satisfy the higher integrality condition?


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                  Problem 1.2.

                  1. (a) Is there a closed formula for the Ehrhart polynomial of order polytopes of fence posets? Further, are these Ehrhart polynomials Ehrhart positive?
                  2. (b) Is the $h^\ast$-polynomial of the order polytopes from (a) real-rooted?
                  3. (c) Can this be extended to generalized permutahedra?


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                      Problem 1.25.

                      Does a rational polygon $P\subseteq \R^2$ with quasi-period 1 satisfy Scott’s inequality? That is, assuming $i\geq 1$, is the inequality $b\leq 2i+7$ satisfied, where $b$ and $i$ represent the number of boundary and interior lattice points of $P$, respectively.


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                          Problem 1.3.

                          Let $ASM_n$ denote the $n$th alternating sign matrix polytope (defined as the convex hull in $\mathbb{R}^{n^2}$ of the $n\times n$ alternating sign matrices). It is known that $i(ASM_n;t)$ is Ehrhart positive. What further can be said about $i(ASM_n;t)$? Furthermore, what can be said about $h^*(ASM_n;z)$?

                            •     Definition: Given two lattice polytopes $P\subseteq \R^d$ and $P'\subseteq\R^{d'}$, the join of $P$ and $P'$ is given by $$P*P' := \mathrm{conv}\{(P,0_{d'},1) \cup (0_{d},P',0)\}.$$

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                              Problem 1.35.

                              For lattice polytopes $P$ and $P'$, which Ehrhart-theoretic properties pass through their join? For example, if $P$ and $P'$ are both Ehrhart positive, is $P*P'$ as well?


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                                  Problem 1.4.

                                  In types B, C, and D, the stretched Clebsch-Gordon coefficients are period-2 quasi-polynomials. Are the coefficients of the constituent polynomials always positive?


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                                      Problem 1.45.

                                      Consider rational polygons of denominator 2. The Ehrhart quasi-polynomial has two constituent polynomials, one of which is an Ehrhart polynomial. Classify the other constituent polynomial.

                                      Note that this is settled for up to 2 interior lattice points.


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                                          Problem 1.5.

                                          Characterize Ehrhart polynomials of tropical polytopes.


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                                              Problem 1.55.

                                              Study the Ehrhart polynomial of Harmonic polytopes and/or bipermutahedra.


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                                                  Problem 1.6.

                                                  1. (a) Consider the Ehrhart polynomial of a lattice polytope $P$. Below are two well-known representations of $i(P;t)$: $$i(P;t) = \sum_{i=1}^d c_it^i = \sum_{i=0}^d h_i^* \binom{t+d-i}{d}.$$ Fix $k\in \Z_{>0}$ and write $i(P;t)$ in the basis $\{(t-k)^i\}_i$. (For example, if $\deg(h^*(P;z)) = s$, then $i(P;t)$ is nonnegative in the basis $\{(t-s+1)^i\}_{i=0}^{d}$.)

                                                    Investigate what happens when changing $k$ from 0 to 1 for non-Ehrhart positive polytopes (or other $k$).
                                                  2. (b) If $P$ is Ehrhart positive in the standard basis, is it Ehrhart positive in the basis $\{t^i(t+1)^{d-i}\}$?

                                                      Cite this as: AimPL: Ehrhart polynomials: inequalities and extremal constructions, available at http://aimpl.org/ehrhartineq.