2 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 $E$ be an $n$-element set and let $\tau$ be an arbor on $E$. Its $h$-polynomial is $h(\tau,t)=\sum_{i=0}^n h_i(\tau)t^i$, where $h_i(\tau)$ counts the lattice points of the arb…