96 problems

For every $d$-dimensional reflexive lattice polytope $P\subset \mathbb{R}^d$, is the Ehrhart polynomial of its polar dual…

For the Hermite normal form simplices $S_{\boldsymbol a}=\operatorname{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol a)$, where $\boldsymbol a=(a_1,\ldots,a_{d-1},N)$, and…

Let $P$ be a $d$-dimensional lattice simplex with the integer decomposition property: for every integer $m\ge 1$ and every $x\in mP\cap\mathbb{Z}^d$, there exist…

Let $P\subseteq\mathbb{R}^d$ be a $d$-dimensional lattice polytope with the integer decomposition property: for every integer $k\geq 1$ and every $x\in kP\cap\mathbb{Z}^d$, there e…

Let $p$ be a prime number and define … Let $E_n$ denote the Euler number, equivalently the number of linear extensions of the zigzag poset of size $n$. Watanabe–Yoshida conjecture.…

Let $M$ be a connected matroid, and let $P_M$ be its base polytope. Say that $P_M$ admits a series-parallel subdivision if it can be subdivided into base polytopes of series-parall…

Let $P$ be a lattice polytope with the integer decomposition property (IDP), meaning that every lattice point in $mP$ is a sum of $m$ lattice points in $P$. Let $h^*_P(x)$ be its…

Let $P$ be a lattice polytope. It is Gorenstein if its associated Ehrhart ring is Gorenstein, and it has the integer decomposition property (IDP) if every lattice point in $mP$ is…

Let $\mathcal{P}$ be a lattice polytope. It is Gorenstein if some positive integer dilate of $\mathcal{P}$ is reflexive, and it has the integer decomposition property if every latt…

Mulmuley's conjecture. The generating function $\tilde{\mathsf{A}}_{\mu[\nu]}^\lambda(t)$ is the Ehrhart series of a polytope.

Unimodality conjecture. The $h^*$-vector of $P_M(W_n)$ is unimodal for every $n\geq 4$.

Castillo–Liu conjecture. Every integral generalized permutohedron $\mathscr{P}$ is Ehrhart positive.

De Loera–Haws–Köppe conjecture. The basis polytope $\mathscr{P}(M)$ of every matroid $M$ is Ehrhart positive.

Let $F_q$ be the $q$th Fibonacci number, let $[F_q]=\{1,2,\ldots,F_q\}$, and let $u_{\mathbb Q}(q;n)$ be the counting quasipolynomial for $q$ queens. Kotěšovec's conjecture. The pe…

Kirillov's unimodality conjecture. For any $n\geq 1$, the Ehrhart $\delta$-polynomial $\delta(\mathcal{P}_n;t)$ is unimodal.

Let $P$ be a $d$-dimensional lattice polytope, and let $h^*(rP)$ denote the $h^*$-polynomial of its dilation $rP$. Beck–Stapledon conjecture. The polynomial $h^*(rP)$ has only dist…

Volume lower-bound conjecture. Every $P\in\mathcal{P}_o^n$ satisfies

Let $\Delta$ be an $n$-dimensional lattice polytope whose $h^*$-polynomial is quadratic, written as … Quadratic $h^*$-polynomial conjecture. One has … This is proposed as a more pr…

Let $\tau$ be a preorder of size $n$, and let $\mathbb E_\tau(u,v)$ be the double Ehrhart polynomial counting lattice points of the two-parameter polytope $\mathcal Q_\tau(r,s)$. B…

Let $\tau$ be a preorder and let $\mathcal Q_\tau$ be its preorder polytope. Magic-positivity conjecture. The Ehrhart polynomial $\operatorname{Ehr}(\mathcal Q_\tau,t)$ is magic po…

Let $E$ be a totally ordered $n$-element set, let $\tau$ be a preorder on $E$, and let $\operatorname{des}(w)$ be the number of descents of a word $w\in E^n$. Define…

Let $\tau$ be a preorder, and for $k,\ell\in\mathbb N$ let $\nabla_\tau(k,\ell)$ denote the number of $k$-multichains in the poset of lattice points of $\ell\mathcal Q_\tau$. Trans…

Let $\sigma$ be an arbor, and let $\mathcal Q_\sigma$ be its arbor polytope. Cha's real-rootedness conjecture. All roots of the Ehrhart polynomial of $\mathcal Q_\sigma$ are real a…

A positroid is a matroid arising from a cell of the totally positive Grassmannian. A matroid base polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are n…

Let $\tau$ be an arbor of size $n$, let $\mathcal Q_\tau$ be its associated arbor polytope, and let $\mathcal D(v)$ denote the union of the blocks at the descendants of a vertex…