On the (co)homology of the poset of weighted partitions

We consider the poset of weighted partitions \Pi _n^w provide a generalization of the lattice \Pi _n. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on node set \{1,2,\dots ,n\} \mathfrak{S}_n has a nice factorization analogous to that of .