I want to know how to determine the number of chains and antichains of a poset from a Hasse-Diagram. Let's say that we have the following Hasse-Diagram:
Let's say that I want to know the number of chains and antichains which contain a minimum of $3$ elements. Of course, I could do it by hand (in this case there are $4$ chains and $7$ antichains) but I'm wondering if there is a way to do it more efficiently which I can also apply to a poset with a higher number of elements and relations.