 This polynomial encodes several pieces of information about the polytope including its volume (the leading coefficient of the polynomial and its number of vertices (the sum of coefficients).By Birkhoff's representation theorem for finite distributive lattices, the upper sets of any partially ordered set form a finite distributive lattice, and every finite distributive lattice can be represented in this way.The upper sets correspond to the vertices of the order polytope, so the mapping from upper sets to vertices provides a geometric representation of any finite distributive lattice.