Derangements and tensor powers of adjoint modules for sl_n

We obtain the decomposition of the tensor space ${\mathfrak{s}\mathfrak{l}}_n^{\otimes k}$ as a module for ${\mathfrak{s}\mathfrak{l}}_n$, find an explicit formula for the multiplicities of its irreducible summands, and (when $n\ge 2k$) describe the centralizer algebra $C=En{d}_{{\mathfrak{s}\mathfrak{l}}_n}({\mathfrak{s}\mathfrak{l}}_n^{\otimes k})$ 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.