39 problems

Chapoton's conjecture. These polynomials are related by the invertible transformation

Let $d$ be the dimension of the classical associahedron. Sleator–Tarjan–Thurston conjecture. Its diameter is equal to $2d-4$ as soon as $d>9$. This conjecture concerns the exact di…

Fabila-Monroy–Flores-Penaloza–Huemer–Hurtado–Urrutia–Wood conjecture. The chromatic number satisfies

Let ${\mathscr A}$ be an orientation of a Coxeter graph of type $A$ or $B$, and let the associated cambrian fan be obtained by gluing maximal cones of the corresponding Coxeter fan…

The transferred operations $\nu$, $\varphi$, $\psi$, and $H$ are sums of monomials in the initial data $\mu_2,\mu_3,\ldots,f,g,h$, with coefficients $\pm1$. Let $K_n$, $n\geq2$, be…

In ite{fz-Ysys}, a complete simplicial fan was associated to an arbitrary finite root system. It was conjectured that this fan is the normal fan of a simple convex polytope (a gene…

Let $K^{\underline{k}}_{\underline{l}}$ and $J^{\underline{k}}_{\underline{l}}$ be the newly constructed manifolds with boundary, called respectively forest biassociahedra and fore…

For integers $m_1,\ldots,m_t\geq 1$, consider the product of simplices … Let $\Asso[m_i]$ denote Loday's associahedron of dimension $m_i$, and let $\star$ denote the shuffle operat…

Let $\mathsf{P}$ be a product of two simplices, and let its pivot polytope be the max-slope pivot rule polytope associated with $\mathsf{P}$ and a generic objective and weight. A p…

A 2-associahedron is one of the posets introduced by the second author as higher analogues of associahedra; these posets form a relative 2-operad and are related to compactified mo…

Let $G$ be the path on $n-2$ vertices, so that the graph associahedron $\mathcal{A}(G)$ is the classical associahedron, and let its chromatic number be denoted by…

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…

A 2-associahedron is the poset constructed by Bottman to capture the behavior of ordered particles on parallel lines in the plane without colinearities. Bottman's polytopality conj…

Let the $s$-associahedron be the polytopal complex associated with the $s$-Tamari lattice and the pure intervals described in the source. Realization conjecture for s-associahedra.…

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 $\mathcal{M}^+_{0,2k}$ be the positive moduli space, let $\overline{\mathcal{M}}^+_{0,2k}$ be its compactification, and let $\mathcal{O}_k$ be the orthogonal momentum amplituhe…

Let $\mathcal{M}^+_{0,2k}=\{\sigma_1<\sigma_2<\cdots<\sigma_{2k}\}/\mathrm{SL}(2,\mathbb{R})$ be the positive moduli space, and let $\overline{\mathcal{M}}^+_{0,2k}$ be its Deligne…

PK-facet conjecture. For all $2\leq k\leq n-2$, the facets of $\mathbb{K}^{(k)}_{n-k}$ are exactly the codimension-one faces minimizing $\gamma_I$ as $I$ ranges over all nonfrozen…

Binary-relations conjecture. For every $J\in\binom{[n]}{3}^{nf}$,

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…

Let $m$ denote the parameter indexing the associahedron, and consider toric statistical models supported on the associahedron. The minimal ML degree conjecture asserts that the num…

Smallest-number-of-solutions conjecture. The number $(n-3)!$ is the smallest possible number of saddle points among all stringy canonical forms reproducing the ABHY associahedron.

Minimal saddle-point conjecture. Among all stringy canonical forms for the ABHY associahedron, ${\bf I}_n$ has the smallest number of saddle points, namely

Minimal saddle-point conjecture. The same pushforward phenomenon holds for generalized associahedra.