28 problems

Let $\Phi$ be a crystallographic root system of rank $\ell$, let $m\geq 1$, and let $\Delta^m_+(\Phi)$ be the positive part of the generalized cluster complex, with $N^+(\Phi,m-1)$…

Let $P$ be a poset that is spherical and shellable, and let $T(P)$ denote its Tchebyshev transform. Spherical shellability conjecture. If $P$ is spherical and shellable, then…

Let $\Pi_n$ be the partition lattice, let $S$ be a set of ranks, and let $\Pi_n^S$ denote its rank-selected subposet. Write $\Delta(\Pi_n^S)$ for the order complex of $\Pi_n^S$, le…

Dual EL-shellability conjecture. The poset $\widehat{\operatorname{Cox}(\mu)^K}$ is dual EL-shellable.

Hexagonal line tiling shellability conjecture. For $k=4,5,$ and $6$, the $k$-cut complex of the hexagonal line tiling is shellable.

Let $\mathrm{M}$ be a matroid. A TN-poset is a lower rank-uniform poset whose lower triangular matrix of lower Whitney numbers of the second kind is totally nonnegative, and UMEL-s…

Let $\Delta_{n-1}^{(d)}$ denote the $d$-skeleton of the $(n-1)$-dimensional simplex. A pure $d$-dimensional shellable complex on $n$ vertices is extendably shellable if every shell…

Let the Generalized Upper Bound Theorem mean that, for a simplicial $d$-polytope $P$ and the cyclic $d$-polytope $C(d,n)$, the inequality $f_{d-1}(P) \geq f_{d-1}(C(d,n))$ implies…

Shellability conjecture. The DKK-triangulation $obreakrootn^{textup{LRL}}$ is shellable because both the modified lexicographic ordering and its reverse are shellings.

Shellability conjecture. The intervals in the level Eulerian poset from Section are shellable, and hence their order complexes are homeomorphic to spheres.

Let $\mathbf{a}=(a_1,\ldots,a_n)$ be a weak composition, and let $\mathcal{P}(\mathbb{D}(\mathbf{a}))$ be the Kohnert poset of its key diagram. Pattern characterization conjecture.…

Let $D$ be a diagram, let $\mathcal{P}(D)$ be its Kohnert poset, and let $\mathcal{F}$ be a finite collection of families of subdiagrams. Finite forbidden-subdiagram conjecture. Th…

Simon's conjecture. Every shellable complex is shelling completable.

Let $\textbf{\textsc{omat}}_+$ be the class of ordered matroid complexes with the indicated extension, and let $\textbf{\textsc{sls}}$ be the class of shellable prefix-pure complex…

For integers $1<k\le n$, let $U_{n,k}$ be the uniform matroid and let its independence complex be the simplicial complex whose facets are the bases of $U_{n,k}$. A partial shelling…

Let $M$ be a matroid. A pinned broken line shelling order is a shelling order obtained from a broken-line shelling whose normal vectors lie in the normal cone of one fixed vertex o…

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 $\mathcal{C}$ be the boundary complex of a cubical polytope. A stable line shelling is a line shelling satisfying the stability condition defined through the realization and in…

Exposed-circuit reformulation of Simon's conjecture. Then ${\mathcal C}$ contains an exposed circuit.

Let $G$ be a group generated as a monoid by a conjugation-closed subset $A$, let $\mathfrak{c}\in G$, and let $A_{\mathfrak{c}}$ be the generators appearing in reduced $A$-factoriz…

Let $G$ be a group generated as a monoid by a conjugation-closed subset $A$, let $\mathfrak{c}\in G$, and let $A_{\mathfrak{c}}$ be the generators appearing in reduced $A$-factoriz…

Let $W$ be a Weyl group, let $x\leq w$, and let $\mathfrak w$ be a reduced word for $w$. Let $\lambda_{x,\mathfrak w}$ be the deletion-index set, and let…

Let $\mathfrak{S}_{n,k}^r$ be the set of $r$-stable permutations, let $\Phi(\mathfrak{S}_{n,k}^r)$ be the associated set on which the graded colexicographic order is imposed, and l…

Let $P_n$ be the uncrossing partial order on matchings, and let ${\hat P}_n$ denote its augmented poset. Lexicographic shellability conjecture. The poset ${\hat P}_n$ is lexicograp…

Let $P_1$ be the permutation class denoted by $P_1$ in the source, and let $\pi$ be a permutation in $P_1$. Write $[\textbf{1},\pi]$ for the interval from the one-element permutati…