18 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 $P$ be a balanced simplicial $d$-polytope, meaning that its underlying graph is $d$-colorable. For $-1\leq i\leq d-1$, let $f_i(P)$ be the number of $i$-dimensional faces, with…

Let $M_d$ be the transfer matrix associated with the McMullen correspondence, with entries … A matrix is totally nonnegative if every minor is nonnegative. Bjbfrner's conjecture. A…

Let $n$ be a positive integer, let $H_n(r)$ denote the number of $n\times n$ magic squares with line sums equal to $r$, and write … where $d=n^2-3n+2$ and $h_0=1$, $h_i=h_{d-i}$ fo…

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…

Let $\mathcal{P}_{6,48}$ be the centrally symmetric simplicial $6$-polytope whose antipodal quotient of the boundary gives a triangulation of $\mathbb{R}P^5$. A triangulation is ve…

Let $h=(1,h_1,\ldots,h_e)$ be the $h$-vector of a simplicial polytope. If $h$ is palindromic and unimodal, define its differential sequence by … An $O$-sequence is the Hilbert func…

Let $P$ be any point set in the plane, and let $\mathrm{WC}_P$ denote its wiggly complex. Wiggly complex polytopality conjecture. For any point set $P$ in the plane, the wiggly com…

Let $d\geq 4$, let $2\leq i\leq d/2$, and let $P$ be a simplicial $d$-polytope. Assume that $P$ has no missing faces of dimension at least $d-i+1$. Equivalently, let…

A simplicial polytope is a polytope whose boundary is a simplicial complex, and its face numbers record the number of faces in each dimension. McMullen's conjecture. The face numbe…

Let $P\subseteq \mathbb R^d$ be a simplicial polytope of dimension $d\geq 2k-1$, with $k\geq 2$. Let $G$ be a missing $(k-1)$-face of $\partial P$, and let $F$ be a $(k-1)$-subset…

Stress-sign conjecture. There exists a $(k-2)$-face $F\subset M$ and an affine $k$-stress $\lambda$ on $P$ such that, for every $(k-1)$-face $G=F\cup v_{F,G}$ of $\partial P$, one…

Let $P\subseteq \mathbb R^d$ be a simplicial $d$-polytope. For an affine $k$-stress on $P$, record the signs of the coefficients in its squarefree part on the $(k-1)$-faces; let…

Klee–Nevo–Novik–Zheng conjecture. Then

Let $P$ be a simplicial polytope, and let $\widehat{\gamma}(P)=(\widehat{\gamma}_0(P),\widehat{\gamma}_1(P),\dots,\widehat{\gamma}_d(P))$ be its gamma-vector with respect to the st…

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 $P$ be a centrally symmetric simplicial $d$-polytope. The higher cs lower bound conjecture. If, for some $3\leq r<\left\lfloor\frac d2\right\rfloor$, … then … Here…

McMullen's conjecture. The sequence $(h_0,h_1,\dots,h_d)$ is the h-vector of the boundary of a simplicial $d$-polytope if and only if $h_0=1$,…