Determinant versus Permanent: salvation via generalization? The algebraic complexity of the Fermionant and the Immanant

The fermionant Fermn(x̄) = ∑ σ∈Sn (−k)c(π) ∏n i=1 xi,j can be seen as a generalization of both the permanent (for k = −1) and the determinant (for k = 1). We demonstrate that it is VNP-complete for any rational k , 1. Furthermore it is #P -complete for the same values of k. The immanant is also a generalization of the permanent (for a Young diagram with a single line) and of the determinant (when the Young diagram is a column). We demonstrate that the immanant of any family of Young diagrams with bounded width and at least n boxes at the right of the first column is VNP-complete.