42 problems

Let $S_{d,n}$ denote the $d$-skeleton of the $(n-1)$-simplex. A pure simplicial complex is extendably shellable if every shelling of it can be extended to a shelling of the whole c…

Let $C_n$ be the cycle graph on $n$ vertices, let $C_n^p$ be its $p$-th power, and let $\Delta_3(C_n^p)$ denote its 3-cut complex. Shellability conjecture. The complex…

Let $C_n$ be the cycle graph on $n$ vertices, let $C_n^2$ be its square, and let $\Delta_k(C_n^2)$ denote its $k$-cut complex. Bayer et al.'s conjecture. Motivated by computational…

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

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

Let $[sigma,pi]$ be an interval of the classical permutation pattern poset. A subinterval is a zero split subinterval if it has the zero-split property described for this poset. Sh…

Induced two-edge obstruction conjecture. Every nonshellable simplicial complex contains, as an induced subcomplex, the two disjoint edges $\{a,b\}$ and $\{c,d\}$, where $a,b,c,d$ a…

Fomin–Reading's shellability conjecture. The complex $\Delta^m(\Phi)$ is shellable.

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…

Let the poset of configuration types for a line and two complete flags be ordered by the covering moves (i)--(v), with each cover labelled by its move type. Lexicographic shelling…

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…

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

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…

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…

Shelling-extender conjecture. If

Ridge-chordality conjecture. If the Alexander dual of $\operatorname{Cl}(\Delta)$ is shellable, then $\Delta$ is ridge-chordal.

Let $\mathcal{N\!C}_{N;k}$ be the poset of $k$-indivisible noncrossing partitions, ordered by refinement. Its order complex is obtained by removing its least and greatest elements.…

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