One of the points of combinatorics I never really learned is how to play correctly with Mobius functions. I mean, I can state Mobius inversion for an arbitrary poset if you give me a moment, but it all ends up a bit hard to manipulate.
This is particularly frustrating since I know that there are a certain number of people out there in the world who know all these tricks by heart. One of them should make a cheat sheet for all these identities.
So, here’s a question that might be easy to answer, but that I can’t quite muddle my way through. Let be a ranked poset (in my example, it’s flats of a matroid) with unique minimal element 0 and maximal 1 and Mobius function . Is there any better expression for the sum