I am searching for an interesting combinatorial application of the following poset $P$: Given the complete Graph $K_n$ on $n$ vertices. The elements of the poset $P$ are the forests in $K_n$ (union of trees) on the $n$ vertices and the binary relation is just "subset".For example: $n=6$, $B=12,23,24,56$ $A=12,23,24$, then $A$ is a subset of $B$, so $A< B$. I would like to have an interesting enumeration problem which uses this poset.Do you have any suggestion for me?How can we calculate the Möbius function of this poset? (something like explicit formulas?)
Best regards, CC