  We obtain the decomposition of the tensor space $\mathfrak{sl}_n^{\otimes k}$
as a module for $\mathfrak{sl}_n$, find an explicit formula for the
multiplicities of its irreducible summands, and (when $n \ge 2k$) describe the
centralizer algebra $C=End_{\mathfrak{sl}_n}(\mathfrak{sl}_n^{\otimes k})$ and
its representations. The multiplicities of the irreducible summands are
derangement numbers in several important instances, and the dimension of $C$ is
given by the number of derangements of a set of $2k$ elements.
