34 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…

Characteristic-independence conjecture. An $h$-vector is level in some characteristic if and only if it is level in every characteristic.

Stanley's conjecture. The $h$-vector of $\Delta$ is a pure $O$-sequence.

Zanello's conjecture. If $R/I$ is level of codimension three, then

Let $\tau$ be an arbor on the ground set $[n]:=\{1,2,\dots,n\}$, and let $\mathcal Q_\tau$ be its associated $n$-dimensional arbor polytope. Define…

Let $R$ be a ladder determinantal ring, and let $h(R)=(h_0,h_1,\ldots,h_s)$ be its $h$-vector. A sequence is log-concave if $h_i^2\geq h_{i-1}h_{i+1}$ for every index for which bot…

Let a matroid be a nonempty simplicial complex satisfying the exchange property: whenever $F$ and $G$ are faces with $|F|<|G|$, some $x\in G\setminus F$ has $F\cup\{x\}$ as a face.…

Let $(h_0,\ldots,h_d)$ be a non-negative integer vector, and for each $k=0,1,\ldots,d+1$ let … with $h_i=0$ for $i>d$. Billera–Lee conjecture. If all these difference vectors are…

Stanley's $h$-vector conjecture. For any matroid $M$, there exists such a set $X$ of monomials.

SI-sequence conjecture. Every Gorenstein $h$-vector in codimension $r=4$ is an SI-sequence.

Stanley–Iarrobino conjecture. Every Gorenstein $h$-vector, in every codimension $r$, is an SI-sequence.

Brown–Colbourn–Devitt conjecture. The $h$-vector of $L(r+1,n)$ should be an absolute lower bound for the $h$-vector of $M(G)^\star$ among all connected simple graphs with $r+1$ ver…

Boesch–Satyanarayana–Suffel conjecture. The graph $L(r+1,n)$ should attain the minimum of $R_G(p)$ among all connected simple graphs with $r+1$ vertices and $n$ edges.

Let $\tau$ be a preorder and let $\tau^\ast$ denote its dual preorder. Preorder h-vector duality conjecture. One has … for every preorder $\tau$. The statement substantially genera…

Let $\tau$ be a preorder of size $n$. For $0\leq i\leq n$, let $h_i(\tau)$ count the lattice points of $\mathcal Q_\tau$ with exactly $i$ nonzero coordinates, and write…

The arbor polytope h-vector realization conjecture. For every arbor $t$, there exists a simple polytope $\Phi_t$ such that

Let $E$ be a finite set, let $\mathrm{P}_1$ and $\mathrm{P}_2$ be polymatroids on $E$, and let $B(\mathrm{P}_1)$ and $B(\mathrm{P}_2)$ denote their base polytopes. Suppose that … F…

Let $R$ be a graded Cohen–Macaulay integral domain, and write $h(R)=(h_0,h_1,\ldots,h_s)$ for its $h$-vector. A sequence is log-concave if $h_i^2\geq h_{i-1}h_{i+1}$ for every inde…

Let $G$ be a cycle graph with $\ell\geq 3$, let $R=E_{\mathbb{K}}[\operatorname{STAB}(G)]$ be its Ehrhart ring, and write its h-vector as $(h_0,h_1,\ldots,h_s)$ with $h_s\neq 0$. S…

Let $\mathbb P^n$ be projective $n$-space, and let $\mathbb{S}(\ell_1,\ldots,\ell_r)$ and $\mathbb{S}(m_1,\ldots,m_s)$ denote contact star configurations formed from distinct hyper…

Let $s\le 4$ and let $m_i$ be the multiplicities of $s$ general fat points. For each $i$, take a contact star configuration formed by $m_i+1$ lines tangent to the same conic. The c…

Brylawski's flawlessness conjecture. The $h$-vector of the broken circuit complex of $\operatorname{\mathrm M}$ is flawless.

Hibi's conjecture. The $h$-vector of the independence complex of $\operatorname{\mathrm M}$ is flawless.

Let $\Delta_{n,\ell}$ be the simplicial complex of $\ell$-zero-sumfree subsets, and let $h_1,\ldots,h_d$ be the relevant entries of its $h$-vector. A simplicial complex is pure whe…

Let $\Delta$ be a PS ear-decomposable simplicial complex, and let $h(\Delta)$ denote its $h$-vector. The PS ear-decomposability conjecture. The vector $h(\Delta)$ is a pure…