Exploring Marked Poset Polytopes

Marked chain-order polytopes, marked order polytopes, and marked chain polytopes are fascinating geometric objects that generalize classical order and chain polytopes. They capture many interesting combinatorial properties of marked posets, which are posets equipped with a function that assigns numerical markings to certain elements. For integral markings, marked poset polytopes are indeed lattice polytopes. Remarkably, the marked order, marked chain, and marked chain-order polytopes share identical Ehrhart polynomials. In this talk, we will explore the intriguing properties of these polytopes as lattice polytopes, present a formula for their Ehrhart polynomial, and discuss surprising connections to other combinatorial structures. These insights shed light on the rich interplay between geometry and combinatorics within the theory of marked polytopes.