133 problems

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 reflexive polytope, meaning a lattice polytope whose polar dual is also a lattice polytope. Let $h^*_P(x)$ denote its $h^*$-polynomial. Hibi's conjecture. Every reflex…

Let $P_M$ be the base polytope of a matroid $M$. A lattice polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are nonnegative. De Loera–Haws–Köppe conject…

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…

Let $P$ be a Gorenstein IDP polytope, meaning an IDP lattice polytope whose associated Ehrhart ring is Gorenstein. Let $h^*_P(x)$ denote its $h^*$-polynomial. Hibi–Ohsugi conjectur…

Let $P\subset\mathbb{R}^d$ be a lattice polytope. It has the integer decomposition property (IDP) if every lattice point of $\operatorname{cone}(P)$ is a nonnegative integral combi…

Let $M$ be a connected matroid of rank $k$ on $n$ elements. Let $E_M(t)$ denote its Ehrhart polynomial, and let $M_{\rm min}$ and $U_{k,n}$ denote respectively the minimal matroid…

Let $u_\mathbb{P}(q;n)$ be the counting quasipolynomial for nonattacking configurations of $q$ copies of a two-move rider $\mathbb{P}$ on the dilated unit board, and let…

A Schubitope is a polytope in the class introduced by Monical, Tokcan, and Yong. A polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are nonnegative. Mon…

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 and let $\mathcal Q_\tau$ be its associated arbor polytope. Let ${\rm Ehr}(\mathcal Q_\tau,t)$ denote its Ehrhart polynomial, and let…

Let $D$ be a decorated ordered set partition (DOSP), and let $w(D)\in\{0,1,\dots,n-1\}$ be its winding number. A DOSP is hypersimplicial if, writing it as…

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

Let $M$ be a matroid and let $\mathscr{P}(M)$ be its matroid polytope. If $\mathscr{P}$ is a lattice polytope of dimension $d$, its $h^*$-polynomial $h^*(\mathscr{P},x)$ is defined…

Castillo–Liu conjecture. Every integral generalized permutohedron $\mathscr{P}$ 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 $P_n$ be the integral polytope of real $n\times n$ doubly stochastic matrices, and let $h_{P_n}^*(x)$ be its Ehrhart $h^*$-polynomial. Stanley's conjecture. The polynomial…

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…

Volume-boundedness conjecture. The value $h^*(1)=\operatorname{Vol}_{\mathbb N}(\Delta)$ is bounded by a constant depending only on the leading coefficient $h_d^*$ of $h^*$.

Stretching conjecture. The coefficients of each polynomial $f_i$ are all nonnegative.

Lee–Vindas-Meléndez–Wang conjecture. The Ehrhart $h^*$-polynomial $h^*(\mathcal{O}(P),t)$ is real-rooted. This conjecture concerns the root structure of Ehrhart $h^*$-polynomials f…

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, let $\mathcal R_\tau$ be the translated reflexive preorder polytope, and let $\mathcal R_\tau^\vee$ be its polar dual. Polar h-star duality conjecture. ……