6 problems

Let ${\mathcal{P}}(m,n)$ be the partial permutohedron, let $t$ be the dilation parameter, and assume $n\geq m-1$. Explicit Ehrhart-polynomial conjecture. Its Ehrhart polynomial is…

Let ${\mathcal{P}}(m,n)$ be the partial permutohedron, and let $v(m,n)$ denote its normalized volume. For each relevant $n$ and $i$, let $p_{n,i}(m)$ be a polynomial in $m$. Polyno…

Let ${\mathcal{P}}(m,2)$ be the partial permutohedron, and write $v(m,2)$ for its normalized volume. Heuer–Striker's volume conjecture. The normalized volume is … The conjecture wa…

Let ${\mathcal{P}}(m,n)$ be the partial permutohedron, and let $k$ be a nonnegative integer. Heuer and Striker proved that, for any $m$ and $k$, the faces of ${\mathcal{P}}(m,m)$ o…

Let $\mathcal{P}(m,n)$ denote the partial permutohedron. Volume conjecture. The normalized volume of $\mathcal{P}(m,2)$ is equal to $3^m-m$. This conjecture gives a closed formula…

Let $\mathcal{P}(m,n)$ be the partial permutohedron and let $\mathcal{B}_m$ be the Boolean lattice of subsets of $[m]$. For a chain in $\mathcal{B}_m$, the difference between its l…