Representations of symmetric groups with non-trivial determinant

We give a closed formula for the number of partitions λ of n such that the corresponding irreducible representation Vλ of Sn has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the 2-core towers of such partitions. We also obtain a formula for the number of partitions of n such that the associated permutation representation of Sn has non-trivial determinant.