Combinatorics

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.


Curriculum:

  1. Basic combinatorial objects and their enumeration.
  2. Power series and generating functions. Examples. Recurrence relations.
  3. Lagrange inversion formula and its applications.
  4. Partitions, Young diagrams. Euler pentagonal theorem.
  5. Asymptotic analysis of combinatorial quantities.
  6. Graphs and their enumeration. Cayley’s formula for the number of trees.
  7. Inclusion–exclusion principle. Möbius inversion formula. Applications.
  8. Surfaces and graphs embedded into surfaces. Enumeration problems for embedded graphs.

Textbooks

  • S. K. Lando, Lectures on generating functions.
  • R. Graham, D. Knuth, O. Patashnik, Concrete mathematics.