Homotopy of Non-Modular Partitions and the Whitehouse Module

We present a class of subposets of the partition lattice 5n with the following property: The order complex is homotopy equivalent to the order complex of 5n−1, and the Sn-module structure of the homology coincides with a recently discovered lifting of the Sn−1-action on the homology of5n−1. This is the Whitehouse representation on Robinson’s space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al. One example is the subposet Pn−1 n of the lattice of set partitions 5n, obtained by removing all elements with a unique nontrivial block. More generally, for 2 ≤ k ≤ n − 1, let Qn denote the subposet of the partition lattice 5n obtained by removing all elements with a unique nontrivial block of size equal to k, and let Pk n = ⋂k i=2 Qn . We show that Pk n is Cohen-Macaulay, and that P k n and Q k n are both homotopy equivalent to a wedge of spheres of dimension (n − 4), with Betti number (n − 1)! n−k k . The posets Qn are neither shellable nor Cohen-Macaulay. We show that the Sn-module structure of the homology generalises the Whitehouse module in a simple way. We also present a short proof of the well-known result that rank-selection in a poset preserves the CohenMacaulay property.