On the topology of two partition posets with forbidden block sizes

We study two subposets of the partition lattice obtained by restricting block sizes. The first consists of set partitions of {1,…,n} with block size at most k, for k≤n−2. We show that the order complex has the homotopy type of a wedge of spheres, in the cases 2k+2≥n and n=3k+2. For 2k+2>n, the posets in fact have the same Sn−1-homotopy type as the order complex of Πn−1, and the Sn-homology representation is the “tree representation” of Robinson and Whitehouse. We present similar results for the subposet of Πn in which a unique block size k≥3 is forbidden. For 2k≥n, the order complex has the homotopy type of a wedge of (n−4)-spheres. The homology representation of Sn can be simply described in terms of the Whitehouse lifting of the homology representation of Πn−1.