Combinatorial Enumeration

Combinatorial Enumeration is a second-year course at University of Amsterdam, offered in periods 4-5 of Semester 2. It is an introduction to generating functions for undergraduate students.

We are interested in combinatorial enumeration; we solve the prob- lem of counting various kinds of combinatorial objects. We do this by making a calculus for enumeration problems. We will associate combinatorial objects to variables and other objects from calculus.

Students will learn the basics of working with generating functions, including using formal power series to count combinatorial objects and solving recurrence equations. They will see many examples of formal power series in such classical settings like the enumeration of compositions of integers and binary strings. Other topics will include binomial coefficients, linear recurrence relations, Catalan numbers, compositions and partitions of integers, enumeration of strings and Stirling numbers. They will then see application of these tools to in more advanced topics, like Young tableaux. Finally, they will see an algebraic extension with some q-theory.

This course was first offered in 2022; here are my course notes from 2022.

Graph Symmetries and Combinatorial Designs

This was a MasterMath course I developed in Winter 2022.

This course brings together two important and interconnected areas of discrete mathematics; algebraic graph theory and design theory.

In the first part of the course, we will cover symmetries of graph and eigenvalue techniques in graph theory. Topics will include vertex-transitivity, Cayley graphs, automorphism groups, eigenvalue interlacing and strongly regular graphs. In the second part of the course, we will study combinatorial designs, including block designs, symmetric designs, Hadamard Matrices, projective geometries, Latin squares and t-designs. We will study their constructions and point graphs, which will give some examples of strongly regular graphs, allowing us to applying techniques from the first half of the course.

We will also give insight into current research by spending two lectures on the application of graph symmetries and design theory in the study various matrices arising from quantum information.

Here are the course notes.