On the lattice of weighted partitions

We introduce and study the lattice of generalized partitions, called weighted partitions. This lattice possesses similar properties of the lattice of partitions. By use of the pictorial representation of a weighted partition, the total number is given by the successive Stirling transforms of the Stirling number of the second kind. We construct an explicit $EL$-labeling on the lattice, which implies this lattice is $EL$-shellable and hence shellable. We compute the M\"obius function and the characteristic polynomial by use of a pictorial representation of a maximal decreasing chain. Further, a maximal decreasing chain is shown to be bijective to a labeled rooted complete binary tree.