11 problems

First signed path conjecture. For every pair of finite, binary trees $(D,R)$ with the same number of leaves, there is a sign assignment of $D$ and a word $w$ of rotation symbols va…

Let $\cG_N$ be the associahedron graph, let $S(\cG_N)$ be the set of lengths of cycles in $\cG_N$, and let $\cD_{N,3}$ denote the comparison set used in the source. Cycle-spectrum…

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…

Residue reduction conjecture. The $(k-2)(n-k-2)$-dimensional residue of $m^{(k)}_n$ on the subspace where $\eta_J=0$ for every $\eta_J\in\text{base}_{k,n}\cup\text{comb}_{k,n}$ can…

Let $\beta$ be the $(n,2)$ torus braid, let $\Delta$ denote the positive half-twist, and consider the mutation graph whose vertices are equivalence classes of Demazure weaves…

Kalai's conjecture. Every intersecting family $\mathcal F$ has at most as many elements as there are triangulations of the $(n-1)$-gon.

Associahedron conjecture. All $\widetilde{\sf M}$-associahedra of a given finite type have the same combinatorial type, meaning that they have the same face lattice. In particular,…

Second signed path conjecture. For every pair $(D,R)$ of finite, binary trees with the same number of leaves, there is a sign consistent path from $D$ to $R$.

Eliahou–Kryuchkov conjecture. For any pair of vertices on $\mathcal{A}_{n-2}$, there exists a path connecting them that can be lifted to a path on the graph $\mathcal{A}_{n-2}^{s}$…

Eliahou–Kryuchkov conjecture. For any pair of vertices on $\mathcal{A}_{n-2}$ there exists a path connecting them that can be lifted to a path on the graph $\mathcal{A}_{n-2}^{s}$.

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