The topology of restricted partition posets

For each composition c→ we show that the order complex of the poset of pointed set partitions π _c→ ^• is a wedge of β(c→) spheres of the same dimensions, where β(c→) is the number of permutations with descent composition c→. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module S ^B where B is a border strip associated to the composition c→. We also study the filter of pointed set partitions generated by a knapsack integer partitions and show the analogous results on homotopy type and action on the top homology.