2 problems

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…