16 problems

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 $\omega_n$ 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…

Optimality conjecture. This partition is also optimal for the multiplicity function.

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

Akitaya–Cardinal–Felsner–Kleist–Lauff's 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.

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…

Fix $n$, and consider the Laplacian eigenvalues of the permutahedron, whose eigenspaces are indexed by Young-diagram shapes of $n$. Call a shape a hook shape when it has the form…

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

Let $n\geq 2$, let $\delta_n=(n-2,\dots,1,0,0)$, and let $P_{\delta_n}\subset\mathbb{R}^n$ be the convex hull of the $S_n$-orbit of $\delta_n$. Write $\gamma_n$ for the permutation…

Embed each permutation $\tau\in S_n$ as $\overline{\tau}=(\tau(1),\ldots,\tau(n))\in\mathbb{R}^n$. These points lie on the sphere … where … and … Embedding the edges of the sorting…

Let a family of realizations of the associahedron be given, each obtained naturally from the classical permutahedron, and let the center of gravity of each realization be defined i…

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