3 problems

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 $\mathcal{P}(m,n)$ 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…