
Mathematics 2015
A $q$enumeration of generalized plane partitionsAbstract: 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 semiregular 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.
