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

Abstract

The fermionant can be seen as a generalization of both the permanent (for $k=-1$) and the determinant. We demonstrate that it is VNP-complete for most cases. Furthermore it is #P-complete for the cases. 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.