Compact Enumeration Formulas for Generalized Partitions

Counting the number of elements in finite sets Si (where i typically ranges over some index set I such as the non–negative integers or a cartesian product of the non–negative integers) is surely one of the oldest and most fundamental problems in mathematics. It is in the nature of the subject that only a few enumeration problems have a compact solution in terms of a simple explicit formula in i which (for instance) only makes use of the basic arithmetic operations and factorials. More surprisingly, combinatorialists can still hardly predict when this rather rare event that an enumeration problem has a nice and elegant formula occurs.