Enumeration of plane partitions

Using some recent results involving Young tableaux and matrices of non-negative integers [10], it is possible to enumerate various classes of plane partitions by actual construction. One of the results is a simple proof of MacMahon's [12] generating function for plane partitions. Previous results of this type [12, 4, 3, 8, 7] involved complicated algebraic methods which did not reveal any intrinsic “reason” why the corresponding generating functions have such a simple form.