Filters in the partition lattice

Abstract

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