18 problems

Let $C_1,C_2\in\mathbb{C}^{d\times d}$ be scaling matrices, with associated delta-matroids, and let $c_1,c_2$ be the corresponding scalings. Delta-matroid invariance conjecture. If…

Let $T \in \operatorname{SSYT}(2,[n])$ have $m$ columns. Let $(S_1,\ldots,S_m)$ be the unique unordered $m$-tuple of pairwise weakly separated one-column tableaux whose union is…

Let $T \in \operatorname{SSYT}(k,[n])$ be a tableau with no frozen factors. Write $T=S^{\cup r}$ with $r$ maximal, where $S=\underline{T}$ has no further common frozen factor. A po…

For $n\geq 3$, let $\Psi_{B_n}(t)$ denote the Ehrhart $h^*$-polynomial of the type $B_n$ hypersimplex, and let $E_{D_n}(t)$ denote the type $D_n$ Eulerian polynomial. Type B recipr…

Let $\Sigma(r,n)$ be the hypersimplex, and consider a subdivision of $\Sigma(r,n)$ into matroid base polytopes. An interior face is a face of the subdivision not contained in the b…

Let $P$ be a $G$-invariant polytope, and suppose that $H^*(P;G)[t]$ is an effective polynomial. Stapledon's positive-coefficient conjecture. If the coefficient of $t^m$ in the ordi…

Let $n$ be any positive integer and let $A_0$ be an alcove in the hypersimplex $\Delta_{2,n}$. Let $\mathcal{P}_{A_0}=(V,E)$ be the breadth-first search order of $\Gamma_{2,n}$ root…

Let $\mathcal{S}$ be a subdivision of $\Delta_{k,n}$ into smaller matroid polytopes. For each $1\leq i\leq n$, let $f_i$ be the number of cells of $\mathcal{S}$ of dimension $n-i$…

Eight-face conjecture. The facet of $\mathbf{N}_{3,n}$ minimizing $\gamma_{j_1,j_2,j_3}$ has exactly eight codimension-$3$ faces combinatorially isomorphic to

A hypersimplex is the matroid polytope of a uniform matroid, and its dual polytope is the polytope dual to that hypersimplex. A shelling is extendable if every partial shelling can…

Let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+1,n}$, and let $\{S_{\hat{\pi}}\}$ be its T-dual collection. Good T-duality correspondence conjecture. The collection…

Let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+1,n}$, and let $\{S_{\hat{\pi}}\}$ be its T-dual collection in $Gr^+_{k,n}$. T-duality conjecture. The collection…

Let $\Delta_{k,n}$ be the hypersimplex, and let $e_J\in\Delta_{k,n}$ range over its $\binom{n}{k}-n$ nonfrozen vertices. For each such vertex, let $\rho_J$ be the corresponding fun…

Early's conjecture. The number of hypersimplicial decorated ordered set partitions of type $(k,n)$ with winding number $d$ is $h^{*}_d(\Delta_{k,n})$.

Generic maximal-rank conjecture. The combinatorial hypersimplices of nonnegative rank $2n$ form a dense open subset of $\mathcal{R}_{n,k}$.

For positive integers $n$ and $r$, let $\Delta_{n,2}^{\operatorname{stab}(r)}$ denote the $r$-stable hypersimplex. Braun–Solus' conjecture. The $h^*$-vector of…

Let $\Delta_{n,2}^{\operatorname{stab}(r)}$ denote the $r$-stable second hypersimplex, and let its Ehrhart $h^\ast$-polynomial be the polynomial whose coefficient sequence is the…

Let $d$ and $n$ be integers with $1\leq d<n$, and let $\Delta(d,n)$ be the convex hull in $\mathbb{R}^n$ of the vectors $\mathbf e_{i_1}+\cdots+\mathbf e_{i_d}$ with…