This course is intended as an elementary introduction to modern combinatorics, with special attention paid to its applications in low-dimensional topology and field theory.
Prerequisites
Real Analysis (Foundations of Analysis, Analysis I, Analysis II: Differentiation and Integration from the list of our courses), and Linear Algebra (Advanced Linear Algebra from the list of our courses).
Curriculum:
- Basic combinatorial objects and their enumeration.
- Power series and generating functions: examples and recurrence relations.
- Lagrange inversion formula and its applications.
- Partitions and Young diagrams; Euler’s pentagonal theorem.
- Asymptotic analysis of combinatorial quantities.
- Graphs and their enumeration: Cayley’s formula for the number of trees.
- Inclusion–exclusion principle and Möbius inversion formula: applications.
- Surfaces and graphs embedded in surfaces: enumeration problems for embedded graphs.
Textbooks
- S. K. Lando, Lectures on Generating Functions.
- R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics.