20 problems

Chan–Robbins–Yuen conjecture. The volume of $\mathcal{F}_{K_n}$ is given by the product above.

Dominance conjecture.

Weak-order lattice conjecture. The weak order of $(G,F)$ always has the structure of a lattice.

Let $G$ be a directed acyclic graph with netflow vector $\mathbf{a}$, and let $(\hat G,\hat{\mathbf{a}},\hat{\mathfrak F})$ be a well-ordered framed augmentation of…

Let $G$ be a strongly planar directed acyclic graph, and let $\mathcal L$ be its framing poset. If $x$ and $y$ cover a common element $z$ and there exists a least upper bound…

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

Flow-statistics conjecture. The $h^*$-polynomial of $\mathcal{F}_{\widehat{\mathop{\textup{Zig}}}_{n}}(\mathbf{u})$ satisfies

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

Yip's conjecture. For $\bm{N}=(1,1,\ldots,1,-n)$, we have

Let $\textup{Car}_{n+1}$ be the Caracol graph on $n+1$ vertices, and let $E_{\textup{Car}_{n+1}}(x)$ denote the Ehrhart polynomial associated with its flow polytope. Caracol Ehrhar…

Let $\textup{Car}_{n+1}$ be the Caracol graph on $n+1$ vertices, and let $\mathcal{F}_{\textup{Car}_{n+1}}(a,b,c,\ldots,c)$ denote its flow polytope with the indicated net flow vec…

Let $\textup{PS}_{n+1}$ be the Pitman–Stanley graph on $n+1$ vertices, and let $\mathcal{F}_{\textup{PS}_{n+1}}(1,0,1,0,\ldots,0)$ denote its flow polytope with the indicated net f…

For $\boldsymbol{a}\in\mathbb{N}^n$, let $\operatorname{Tes}_n(\boldsymbol{a})$ be the polytope of upper triangular $n\times n$ matrices with nonnegative entries and hook sum…

Let $K^{C}_{n+1}$ be the complete signed graph with $n+1$ vertices, containing all edges $(i,j,\pm)$ for $1\leq i<j\leq n+1$ and $(i,i,+)$ for $1\leq i\leq n$, and let…

Let $\mathcal{P}_n$ be the poset of connected threshold graphs with vertices $0,1,\ldots,n$, ordered by subgraph inclusion: $H\preceq G$ when $H$ is a subgraph of $G$. For threshol…

Let $G$ be a threshold graph with $n+1$ vertices. The weighted Ehrhart series $\operatorname{Ehr}_{q,t}({\mathcal{F}}_G(-n,1,\ldots,1))$ is a polynomial with nonnegative coefficien…

Mészáros–Morales conjecture.

Let $K_n^D$, $K_n^B$ and $K_n^C$ be the signed complete graphs whose edges correspond to the positive roots of types $D_n$, $B_n$ and $C_n$, respectively. Write…

Let $K_n^D$ be the complete signed graph with vertex set $[n]$, containing all edges $(i,j,\pm)$ with $1\leq i<j\leq n$, and let … be its flow polytope. Let…

Let $CRYC_{n+1}$ and $CRYD_{n+1}$ denote the type $C_{n+1}$ and type $D_{n+1}$ analogues of the Chan–Robbins–Yuen polytope $CRYA_n$. For a nonnegative integer $k$, let … be the $k$…