Some combinatorial topics I have found to be useful in competitive math are
- Burnsides Lemma
- Stars and Bars
- Recursion
- Expected Value(Need help on this one)
- Extremal principle
- PIE
- Combinatoric Geometry
- Indicator functions with Sets
- Partially ordered sets(Erdős–Szekeres,Dilwort's,Mirsky's...)
If you are wondering what level I am looking for I am looking to be able to solve late AIME(around problems 10-15) combinatorics problems easily.