Filters in the partition lattice

Abstract

Given a filter $\Delta$ in the poset of compositions of $n$, we form the filter ${\Pi }_{\Delta }^{\ast }$ in the partition lattice. We determine all the reduced homology groups of the order complex of ${\Pi }_{\Delta }^{\ast }$ 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 $a,a+d,\dots ,a+(a-1)\cdot d$, extending work of Browdy.

29 pages, 1 figures and 2 tables