Algebraic and Geometric Methods in Enumerative Combinatorics

By Federico Ardila. 2014

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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 which we only know how to count using these tools. Our goal in this chapter is to highlight some key aspects of the rich interplay between algebra, discrete geometry, and combinatorics, with an eye towards enumeration.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

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