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 rst consists of set partitions of f1; : : : ; ng 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 S n?1-homotopy type as the order complex of n?1 ; and the S n-homology representation is the \tree representation"of Robinson and Whitehouse. We present similar results for the subposet of n 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 S n can be simply described in terms of the Whitehouse lifting of the homology representation of n?1 : 0. Introduction In this paper we investigate the homology of two subposets of partitions of an n-element set with restricted block sizes, the subposet n;k of the partition lattice n whose block sizes are bounded above by a xed integer k n ? 2; and the subposet n;6 =k of n where the block size k n ? 1 is forbidden. (The (reduced) homology is taken over the rationals for the representation-theoretic results, and over the integers otherwise.) Some of our results were announced in S4]. This work is motivated by a formula in S2] for the representation of the symmetric group S n on the Lefschetz module of any subposet of n obtained by restricting block sizes. For the posets considered in this paper, these formulas show that the symmetric group acts on the Lefschetz module in a surprisingly nice way. For k = 2 the order complex ((n;k) is the \matching complex"of BLVZ]. The rational homology in this case was completely determined in earlier work of Bouc ((Bo]). The poset n;k is also the intersection lattice of a relative arrangement, a concept recently introduced by Welker ((We2]). For 2k + 2 n; the Lefschetz module Alt((n;k); turns out to be plus or minus a true representation of S n ; and for 2k + 2 > n; it is in fact (plus or minus) a lifting of the representation of S n?1 on the homology of the partition lattice