37 problems

Let $\bm M_d$ be the matrix in the McMullen correspondence, of size $\left(\left\lfloor\frac{d}{2}+1\right\rfloor\right)\times(d+1)$. A matrix is totally non-negative if every mino…

Let $p(t)$ be a polynomial with positive integer coefficients and constant term equal to $1$. Bell–Skandera conjecture. The polynomial $p(t)$ is the $f$-polynomial of a simplicial…

Let $Q$ be a cubical $d$-polytope with $n$ vertices, and write $f_k(Q)$ for its number of $k$-dimensional faces. Jockusch's cubical lower bound conjecture. For $1\leq k\leq d-2$, ……

Let $\Delta$ be a doubly Cohen–Macaulay flag simplicial complex of dimension $d-1$, and let $h_\Delta=(h_0,\ldots,h_d)$ be its $h$-vector. Suppose that the equality case of the coe…

Björner–Swartz conjecture. The $g$-vector of any doubly Cohen–Macaulay complex is an $M$-sequence.

Kühn­el's componentwise minimality conjecture. The $f$-vector

$L(3,1)$ $f$-vector conjecture. The $f$-vectors of triangulations of the lens space $L(3,1)$ are characterized by

3-torus $f$-vector conjecture. The $f$-vectors of triangulations of the $3$-torus $T^3$ are characterized by

Let $X$ be a doubly CohenMacaulay, or $2$-CM, complex, and let $g(X)$ denote its $g$-vector. An $M$-sequence is the $f$-vector of an order ideal of monomials. Bjrner and Swartz's…

The $g$-theorem characterizes the $f$-vectors of boundary complexes of simplicial polytopes. For a homology sphere, its $g$-vector is the sequence associated with its $f$-vector, a…

Let $P$ be a convex $d$-polytope with $f$-vector $(f_0,f_1,\ldots,f_{d-1})$, where $f_k$ denotes the number of $k$-dimensional faces. Bjrner's quarter-monotonicity conjecture. The…

Let $P$ be a convex $d$-polytope with $f$-vector … where $f_k$ is the number of $k$-dimensional faces of $P$. The unimodality conjecture. For each $d$-polytope $P$ there is an inte…

Spanning conjecture. The set of $f$-vectors of all ordinary $d$-polytopes spans the Euler hyperplane. A spanning set consists of the ordinary polytopes

Let $f=(f_{-1},f_0,\ldots,f_d)$ be the $f$-vector of a simplicial complex, where $f_i$ denotes the number of $i$-dimensional faces. A sequence is unimodal if it weakly increases an…

Let $Delta$ be a $(d-1)$-dimensional pure flag complex on $2d$ vertices, and let its $f$-vector be the vector counting faces by dimension. Levit–Mandrescu conjecture. The $f$-…

Partition-graph $f$-polynomial conjecture. The approach developed in this paper may be modified to obtain the $f$-polynomials of partition graphs.

Let a matroid be viewed as a pure simplicial complex, and let $(h_0,h_1,\ldots,h_d)$ be the $h$-vector of its simplicial complex. A sequence is a pure $O$-sequence if it is the deg…

Let $P$ be a poset with a partition $P=C\sqcup O$ such that $\hat{0}\in C$ and $\hat{1}\in O$. The associated chain-order polytope is denoted by $\mathcal{O}_{C,O}(P)$. Varying the…

Excess bound conjecture. If $v\notin\mathcal{E}^d$, then

For a three-dimensional polytope $P$, let $f(P)$ be its $f$-vector and let $\operatorname{Symm}(P)=\{A\in\mathbb{R}^{3\times 3}:A\cdot P=P\}$ be its linear symmetry group. For a gr…

Let $F(G)$ denote the set of $f$-vectors realizable by three-dimensional polytopes with symmetry group $G$, and let the certificates used to characterize these sets be those constr…

The $f$-vector of a simplicial polytope is the vector recording the numbers of faces of each dimension. McMullen's $g$-conjecture. The $g$-conjecture posits a complete characteriza…

Let $d$ be a positive integer. For a cubical $d$-polytope $Q$, let $g^c(Q)$ be its cubical $g$-vector, and let $\mathcal{C}_d$ be the minimal closed cone containing all such vector…

Extended Hibi–Li conjecture. Given partitions

Let $\mathcal{F}(\mathcal{P}^4)$ denote the set of $f$-vectors of all $4$-polytopes, and let $\mathbin{{\scriptstyle\boxplus}”}$ be the modified addition realized by connected sum…