3 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…

Let $\Delta_{e_k}$ denote the Delta operator on symmetric functions, and let $e_n$ be the elementary symmetric function of degree $n$. A symmetric function is $q,t$-Schur positive…