Filters in the partition lattice

Given a filter (Delta ) in the poset of compositions of n, we form the filter (Pi ^{*}_{Delta }) in the partition lattice. We determine all the reduced homology groups of the order complex of (Pi ^{*}_{Delta }) as ({mathfrak S}_{n-1})-modules in terms of the reduced homology groups of the simplicial complex (Delta ) 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, ldots , a + (a-1) cdot d), extending work of Browdy.