Determinant versus Permanent: Salvation via Generalization?

摘要

The fermionant Ferm_n~k(x) = ∑_(σ∈S_n)(-k)~(c(π))∏_(i=1)~nx_(i,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~ε boxes at the right of the first column is VNP-complete.