Counting integer points in polytopes associated with directed graphs

We are interested in enumerating the integer points in certain polytopes that are naturally associated with directed graphs. These polytopes generalize Stanley's order polytopes and also (P,ω)(P,ω)-partitions. A classical result states that the number of integer points in any given rational polytope can be expressed by a formula that is piecewise a quasipolynomial in certain parameters of the polytope, and, remarkably, the domains of validity of the involved quasipolynomials overlap. In the case of our special polytopes, the quasipolynomials are shown to be polynomials. We investigate the domains of validity of these polynomials and demonstrate how the overlaps can be used to explore the zero set of the polynomials. We have a closer look at the counting of Gelfand–Tsetlin patterns, which can be phrased as the counting of integer points in a polytope associated with a particular directed graph. We conjecture that the zeros that can be deduced by studying the overlaps essentially determine the enumeration formula in this case.