Another spin-off of the Noga-poster-formula-competition is a MathOverflow question: Important formulas in combinatorics.
The question collects important formulas representing major progress in combinatorics.
So far there are 31 formulas and quite a few were new to me. There are several areas of combinatorics that are not yet represented. As is natural, many formulas come from enumerative combinatorics. Don’t hesitate to contribute (best – on MathOverflow) more formulas!
Here are a few:
Combinatorics and more 
The second expression from a bottom is not a formula.
How would you call it?