Enumerative combinatorics is about counting. The typical question is to find the number of objects with a given set of properties.

However, enumerative combinatorics is not just about counting. In “real life”, when we talk about counting, we imagine lining up a set of objects and counting them off: 1, 2, 3, ... However, families of combinatorial objects do not come to us in a natural linear order. To give a very simple example: we do not count the squares in an m × n rectangular grid linearly. Instead, we use the rectangular structure to understand that the number of squares is m· n. Similarly, to count a more complicated combinatorial set, we usually spend most of our efforts understanding the underlying structure of the individual objects, or of the set itself.

Many combinatorial objects of interest have a rich and interesting algebraic or geometric structure, which often becomes a very powerful tool towards their enumeration. In fact, there are many families of objects that we only know how to count using these tools.