Filters in the partition lattice

Given a filter $Δ$ 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 ${\mathfrak S}_{n-1}$-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 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.

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