counting tricks, Bag of tricks for counting:

• Multiplication principle
• Addition principle
• Complement principle
  • more general: Inclusion-exclusion principle, example
• Multiple counting
• When are things the same
• Pidgeon hole principle
• Counting the same thing in multiple ways and then making them equal, can help count related things. Like in proof of Sperner's lemma, or local LYM inequality

Lecture notes

See Analytic combinatorics

https://en.wikipedia.org/wiki/Latin_square

Some approaches:

• Recursive method (I think the one used in Analytic combinatorics)
• Bijective approach
• Matrix integrals
• Symmetryes

Probabilistic combinatorics

Permutations, Combinations – Generalized permutations and combinations : Stars and bars

Derangement

Combinatorics of Power set

Chain set, Antichain, Shadow set

Combinatorial structure

Many relations to Probability theory

Catalan numbers

Random High-Dimensional Combinatorial Objects

Combinatorics has applications and relations to Kolmogorov complexity, here for instance is a proof of the infinitude of prime numbers using KC

"Combinatorics", Bollobas. "Extremal combinatorics", Jukna