Resumen:

For any pair of positive integers n and k, we introduce a family of posets that generalize both the bond lattice of a graph and the poset of weighted partitions introduced by Dotsenko and Khoroshkin in the case k=2 of the complete graph, which was extended and studied by González D'León and Wachs. We show that for chordal graphs, these posets are EL-shellable and hence Cohen-Macaulay. The Möbius values of the máximal intervals induce an interesting family of symmetric functions that conjecturally distinguish trees. In the case of path graphs, the family of symmetric functions coincides with Garcia-Haiman's parking function symmetric functions. In the case k=2 of a tree, the symmetric function conjecturally contains the information of the h-vectors of graph associahedra studied by Postnikov, Reiner and Williams, and others. This is joint work with Michelle Wachs.