  In this paper, we enumerate parallelogram polycubes according to several
parameters. After establishing a relation between Multiple Zeta Function and
the Dirichlet generating function of parallelogram polyominoes, we generalize
it to the case of parallelogram polycubes. We also give an explicit formula and
an ordinary generating function of parallelogram polycubes according to the
width, length and depth, by characterizing its projections. Then, these results
are generalized to polyhypercubes.
