Filters in the partition lattice

Given a filter Δ in the poset of compositions of n, we form the filter in the partition lattice. We determine all the reduced homology groups of the order complex of as S_{n - 1}-modules in terms of the reduced homology groups of the simplicial complex Δ and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank–Hanlon–Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated with integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression a, a+ d, … , a+ (a- 1) · d, extending work of Browdy.