The *Jack characters* are a one-parameter deformation of the characters of the symmetric group; a deformation given by the coefficients in the expansion of the *Jack polynomials* in the basis of power-sum symmetric functions. For each integer partition of *n*, we give a simple combinatorial formula for the sum of the corresponding Jack character evaluated over all integer partitions of *n* with no singleton parts. As a corollary, we obtain closed forms for the eigenvalues of the so-called *permutation* and *perfect matching derangement graphs*, resolving an open question in algebraic graph theory. 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*.

# Nathan Lindzey – Jack Derangements

Abstract: