Gorenstein Fano polytopes arising from order polytopes and chain polytopes

Richard Stanley introduced the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ arising from a finite partially ordered set $P$, and showed that the Ehrhart polynomial of $\mathcal{O}(P)$ is equal to that of $\mathcal{C}(P)$. In addition, the unimodular equivalence problem of $\mathcal{O}(P)$ and $\mathcal{C}(P)$ was studied by the first author and Nan Li. In the present paper, three integral convex polytopes $\mathrm{\Gamma }(\mathcal{O}(P),-\mathcal{O}(Q))$, $\Gamma($\mathcal{O}$(P), -$\mathcal{C}$(Q))$and$\Gamma($\mathcal{C}$(P), -$\mathcal{C}$(Q))$,where$P$and$Q$arepartiallyorderedsetswith$| P | = | Q |$, will be studied. First, it will be shown that the Ehrhart polynomial of $\Gamma($\mathcal{O}$(P), -$\mathcal{C}$(Q))$coincideswiththatof$\Gamma($\mathcal{C}$(P), -$\mathcal{C}$(Q))$\mathrm{.}Furthermore,when$P$and$Q$ possess a common linear extension, it will be proved that these three convex polytopes have the same Ehrhart polynomial. Second, the problem of characterizing partially ordered sets $P$ and $Q$ for which $\mathrm{\Gamma }(\mathcal{O}(P),-\mathcal{O}(Q))$ or $\mathrm{\Gamma }(\mathcal{O}(P),-\mathcal{C}(Q))$ or $\Gamma($\mathcal{C}$(P), -$\mathcal{C}$(Q))$ is a smooth Fano polytope will be solved. Finally, when these three polytopes are smooth Fano polytopes, the unimodular equivalence problem of these three polytopes will be discussed.