11 problems

A matroid $M$ has an $h$-vector $h(M)=(h_0,ots,h_d)$ determined by the face numbers of its independent-set complex. A vector is a pure $O$-sequence if it is the $F$-vector of a pur…

Chari's shellable $O$-sequence conjecture. The $h$-vector $h(M)$ of every coloop-free matroid is a shellable $O$-sequence.

Let $M$ be a matroid complex, and let $h(M)$ denote its $h$-vector. A finite sequence is a pure $O$-sequence if it is the $h$-vector of a pure order ideal of monomials, equivalentl…

Large-type non-log-concavity conjecture. For every codimension $r\geq 3$ and all sufficiently large types $t$, there exists a pure $O$-sequence of codimension $r$ and type $t$ that…

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…

Let $d>1$ be a fixed integer. A based matroid is a triple $(\Delta,B,<)$ consisting of a rank-$d$ matroid $\Delta$, a basis $B$, and a total order $<$ on $E(\Delta)-B$. For an inde…

Let $A$ be the degree matrix of a standard determinantal scheme, and let $h^A$ denote the associated $h$-vector. Assume that $A$ has no zero entries. Converse conjecture. The vecto…

Let $(1,h_1,\dotsc,h_e)$ be a sequence of natural numbers, and fix an index $i$. Suppose that for some positive integer $\alpha$, both … and … are pure $O$-sequences. Interval conj…

Let $h=(1,h_1,\dotsc,h_e)$ and $h'=(1,h'_1,\dotsc,h'_{e-1})$ be pure $O$-sequences. For a vector $H=(1,H_1,H_2,\dots,H_t)$, write $\Delta H=(1,H_1-1,H_2-H_1,\dots,H_t-H_{t-1})$ for…

Let a pure simplicial complex be one whose facets all have the same dimension, and let its $f$-vector record the number of faces in each dimension. An $f$-vector has the interval p…

Let $(1,h_1,\ldots,h_i,\ldots,h_e)$ be a pure $O$-sequence, and let $\alpha$ be a positive integer such that $(1,h_1,\ldots,h_i+\alpha,\ldots,h_e)$ is also a pure $O$-sequence. Int…