  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 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 $a, a + d, \ldots, a + (a-1) \cdot d$, extending
work of Browdy.
