A basis for the top homology of a generalized partition lattice

For a fixed positive integer k, consider the collection of all affine hyperplanes in n-space given by x(i) - x(j) = m, where i, j is an element of [n], i not equal j, and m is an element of {0, 1,...,k}. Let L-n,L-k be the set of all nonempty affine subspaces (including the empty space) which can be obtained by intersecting some subset of these affine hyperplanes. Now give L-n,L-k a lattice structure by ordering its elements by reverse inclusion. The symmetric group G(n) acts naturally on L-n,L-k by permuting the coordinates of the space, and this action extends to an action on the top homology of L-n,L-k. It is easy to show by computing the character of this action that the top homology is isomorphic as an G(n)-module to a direct sum of copies of the regular representation, CG(n). In this paper, we construct an explicit basis for the top homology of L-n,L-k, where the basis elements are indexed by all labelled, rooted, (k + 1)-ary trees on n-vertices in which the root has no 0-child. This construction gives an explicit G(n)-equivariant isomorphism between the top homology of L-n,L-k and a direct sum of copies of CG(n).