  For each integer partition $\lambda \vdash n$ we give a simple combinatorial
expression for the sum of the Jack character $\theta^\lambda_\alpha$ over the
integer partitions of $n$ with no singleton parts. For $\alpha = 1,2$ this
gives closed forms for the eigenvalues of the permutation and perfect matching
derangement graphs, resolving an open question in algebraic graph theory. A
byproduct of the latter is a simple combinatorial formula for the immanants of
the matrix $J-I$ where $J$ is the all-ones matrix, which might be of
independent interest. Our proofs center around a Jack analogue of a hook
product related to Cayley's $\Omega$--process in classical invariant theory,
which we call the principal lower hook product.
