  In 1999, Pitman and Stanley introduced the polytope bearing their name along
with a study of its faces, lattice points, and volume. The Pitman-Stanley
polytope is well-studied due to its connections to probability, parking
functions, the generalized permutahedra, and flow polytopes. Its lattice points
correspond to plane partitions of skew shape with entries 0 and 1. Pitman and
Stanley remarked that their polytope can be generalized so that lattice points
correspond to plane partitions of skew shape with entries $0,1, \ldots , m$.
Since then, this generalization has been untouched. We study this
generalization and show that it can also be realized as a flow polytope of a
grid graph. We give multiple characterizations of its vertices in terms of
plane partitions of skew shape and integer flows. For a fixed skew shape, we
show that the number of vertices of this polytope is a polynomial in $m$ whose
leading term, in certain cases, counts standard Young tableaux of a skew
shifted shape. Moreover, we give formulas for the number of faces, as well as
generating functions for the number of vertices.
