92 problems

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…

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…

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…

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 $G$ be a 2-connected graph and let $ij$ be any edge of $G$. Let $\Gamma$ be a unimodular triangulation of the subcomplex of $P_{G\setminus ij}$ consisting of all facets visible…

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

The bivariate generating function is … A weighted lattice-point cone conjecture. The generating function $A_\mu(z,q)$ is a weighted generating function of lattice points of a disjo…

Let $P$ be an IDP polytope, meaning a lattice polytope such that for every $k\in\mathbb{N}_{\ge 2}$, each lattice point in $kP$ is a sum of $k$ lattice points in $P$. If … is the…

For $n\geq 3$, let $\Psi_{B_n}(t)$ denote the Ehrhart $h^*$-polynomial of the type $B_n$ hypersimplex, and let $E_{D_n}(t)$ denote the type $D_n$ Eulerian polynomial. Type B recipr…

Let $G$ be a graph, let $\mathbb{C}_G$ be its cosmological polytope, and let $T$ be a triangulation of $\mathbb{C}_G$ arising from a good term order on $R_G$. For each simplex…

Let $P_k$, $P_\ell$, and $P_m$ be paths of lengths $k$, $\ell$, and $m$, respectively, and let $G$ be the graph obtained by identifying the three starting nodes and the three endin…

Let $G=(V,E)$ be a graph, and let $h^*(\mathbb{C}_G;z)$ denote the $h^*$-polynomial of its cosmological polytope. For polynomials with nonnegative coefficients, write $f\preceq g$…

Let $i\in\mathbb{Z}_{\geq 2}$ be the number of interior lattice points of a half-integral polygon, and require the polygon to have at least $2$ boundary lattice points. An Ehrhart…

The real-rootedness and log-concavity conjecture. One has

Bóna–Ju–Ann conjecture. If $k=2\ell$, then

Ju's conjecture. The polynomial in the numerator of this Ehrhart series is symmetric, also known as palindromic.

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

Let $P$ be a lattice polytope, and let ${\mathcal H}_P\subseteq\mathbb{R}[y_0,\mathbf{y}]$ be its harmonic algebra, with bigraded ideal $\overline{{\mathcal H}}_P$. Harmonic algebr…