On the number of faces of marked order polytopes

In this paper, we present a new method for computing the f-vector of a marked order polytope. Namely, given an arbitrary (polyhedral) subdivision of an arbitrary convex polytope, we construct a cochain complex (over the two-element field Z_2) such that the dimensions of its cohomology groups equal the components of the f-vector of the original polytope. In the case of a marked order polytope and its well-known cubosimplicial subdivision, this cochain complex can be described purely combinatorially -- which yields the said computation of the f-vector. Of independent interest may be our combinatorial description of the said cubosimplicial subdivision (which was originally constructed geometrically).