This is intended to be an elementary introduction to modern combinatorics with special attention paid to its applications in low-dimensional topology and field theory. No prerequisites.
- Basic combinatorial objects and their enumeration.
- Power series and generating functions. Examples. Recurrence relations.
- Lagrange inversion formula and its applications.
- Partitions, Young diagrams. Euler pentagonal theorem.
- Asymptotic analysis of combinatorial quantities.
- Graphs and their enumeration. Cayley’s formula for the number of trees.
- Inclusion–exclusion principle. Möbius inversion formula. Applications.
- Surfaces and graphs embedded into surfaces. Enumeration problems for embedded graphs.
- S. K. Lando, Lectures on generating functions.
- R. Graham, D. Knuth, O. Patashnik, Concrete mathematics.