  Walks on the representation graph $\mathcal R_{\mathsf{V}}(\mathsf{G})$
determined by a group $\mathsf{G}$ and a $\mathsf{G}$-module $\mathsf{V}$ are
related to the centralizer algebras of the action of $\mathsf{G}$ on the tensor
powers $\mathsf{V}^{\otimes k}$ via Schur-Weyl duality. This paper explores
that connection when the group is $\mathbb{Z}_2^n$ and the module $\mathsf{V}$
is chosen so the representation graph is the $n$-cube. We describe a basis for
the centralizer algebras in terms of labeled partition diagrams. We obtain an
expression for the number of walks by counting certain partitions and determine
the exponential generating functions for the number of walks
