Filters in the partition lattice

TL;DR

This paper investigates the homological properties of filters in the partition lattice derived from filters in the composition poset, generalizing previous results and applying to various combinatorial structures.

Contribution

It determines the reduced homology groups and homotopy types of order complexes of these filters, linking them to Specht modules and extending prior work.

Findings

Homology groups expressed via reduced homology of associated simplicial complexes.

Homotopy type of the order complex characterized.

Results apply to filters from integer knapsack partitions and block size conditions.

Abstract

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.