# 3D lattice paths and the enclosed “volume” beneath

If I select a lattice path uniformly at random from the set of all paths on a rectangular subset of the integer lattice $$[0,n] \times [0,m] \cap \mathbb{Z}^2$$, starting at $$(0,0)$$ and finishing at $$(n,m)$$, I can use the Gaussian binomial coefficient to get a p.m.f. of the area "beneath" the path (i.e. the 2d area enclosed by the path, and the lower and right sides of the rectangle).

In 3d, the analogue of this area has not been described before.

If I select a path from $$(0,0,0)$$ to $$(n,m,k)$$ in the cuboid $$[0,n] \times [0,m] \times [0,k]$$, uniformly at random, how can I describe its enclosed volume, according to the law: the projection of the face of each enclosed cube onto the faces of the cuboid must lie under the projection of the lattice path onto that face, as in the pictures below.