Filters in the partition lattice

Given a filter 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 -modules in terms of the reduced homology groups of the simplicial complex 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 , extending work of Browdy.