6 problems

Let $\tau$ be a preorder of size $n$ and let $h(\tau,t)$ be its $h$-polynomial. Real-rootedness conjecture. The polynomial $h(\tau,t)$ has only real roots for every preorder $\tau$…

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 $\tau^\ast$ denote its dual preorder. Preorder h-vector duality conjecture. One has … for every preorder $\tau$. The statement substantially genera…

Let $\tau$ be a preorder of size $n$. For $0\leq i\leq n$, let $h_i(\tau)$ count the lattice points of $\mathcal Q_\tau$ with exactly $i$ nonzero coordinates, and write…

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…