Filters in the partition lattice

Given a filter Λ in the poset of compositions of n, we form the filter Π_Δ^∗in the partition lattice. We determine all the reduced homology groups of the order complex of Π_Δ^∗as S_n-1-modules in terms of the reduced homology groups of the simplicial complex A 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 to 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.