11 problems

Let $\omega_n$ be an $n$-element uniform sorting network, let ${\mathbb S}_n$ denote the permutahedron, and let $d_\infty$ be the distance from a sorting network to a great circle…

Facet-Hamiltonian conjecture. The $B$-permutahedron of any dimension $d\geq 2$ has a facet-Hamiltonian cycle.

A type $B$ permutahedron is the Coxeter permutahedron associated with the type $B$ reflection group; a polytope is facet-Hamiltonian when its skeleton contains a cycle that visits…

Yip's conjecture. For $\bm{N}=(1,1,\ldots,1,-n)$, we have

Edge-isoperimetric conjecture for the permutahedron.

For each sequence $s$, let the $s$-permutahedron be the generalized permutahedral object defined in the source. A polytopal subdivision is a subdivision whose cells are polytopes.…

Let $s$ be a weak composition with no zero entries except possibly $s(1)$. Removing facets of the $s$-permutahedron. There exists a geometric realization of the $s$-permutahedron s…

Let $s$ be a weak composition. For a pure interval $[T,T+A]$ of the $s$-weak order, consider the poset of pure intervals contained in it and the restriction of the $s$-weak order t…

Ceballos–Pons realization conjecture. The $\operatorname{s}$-permutahedron $\text{Perm}_{\operatorname{s}}$ can be realized as a polyhedral subdivision of a polytope that is combin…

Dominance-order conjecture. If $\nu \vartriangleright \gamma$, then

Vertex-barycentre conjecture. The vertex barycentres of ${\mathsf{Perm}}^{\pmb a}(W)$ and ${\mathsf{Asso}}^{\pmb a}_c(W)$ coincide.