A $q$-enumeration of generalized plane partitions

MacMahon proved a simple product formula for the generating function of the plane partitions fitting in a given rectangular box. The theorem implies the number of lozenge tilings of a semi-regular hexagon on the triangular lattice. By investigating the lozenge tilings of a hexagon with a hole on the boundary, we generalize the ordinary plane partitions to piles of unit cubes fitting in a union of several adjacent rectangular boxes. We extend MacMahon's classical theorem by proving that the generating function of the generalized plane partitions is given by a simple product formula.