10 problems

Let $S(n,d)$ be the set of subsets $S\subset\binom{[n]}{d+1}$ corresponding to triangulations of the cyclic polytope $C(n,d)$. Let $S_1(n,d)$ and $S_2(n,d)$ be the two partial orde…

Reiner's conjecture. If $[I,J]$ is facial, then it is spherical. If $[I,J]$ is not facial, then it is contractible.

Let $w\in \widetilde{\mathfrak{S}}_n$, let $\mathcal{C}_w(n,k)$ denote the relevant collection of consistent subsets, and let $R\in \mathcal{C}_w(n,k)$. Define $G_R$ by including e…

For a positive integer $n$ and an affine permutation $w\in \widetilde{\mathfrak{S}}_n$, consider the higher-Bruhat-type theorem stated immediately before this conjecture, including…

For an affine permutation $w\in \widetilde{\mathfrak{S}}_n$, let $\mathcal{G}(w)$ be the directed graph whose vertices are commutation classes of reduced words for $w$, with edges…

For integers $1 \le k \le n$ and $S \in \mathcal{C}(n,k)$, let $\mathcal{P}_S$ be the consistent poset associated to $S$. Rankedness conjecture. The poset $\mathcal{P}_S$ is ranked…

For positive integers $n$ and $d$, let $\mathbf{SQ}(n,d)$ be the digraph whose vertices are symmetric cubillages of the cyclic zonotope $Z([n],d)$ and whose directed edges are the…

Let $n=2m$ or $n=2m+1$, and let $\mathbf{C}(m,d)$ and $\mathbf{B}(m,d)$ be the posets of symmetric cubillages in the cyclic zonotope $Z(n,d)$. Denote their standard and anti-standa…

Galambos–Reiner's conjecture. The quantity $\mathsf{fb}(m)$ is a tight upper bound on the cardinality of acyclic sets described in terms of higher Bruhat orders.

Let $B(n,d)$ be the higher Bruhat order, and let an interval of $B(n,d)$ be facial if it arises from the set of simple arrangements obtained by resolving the crossings of a non-sim…