In the context of averaging over network paths, I arrived at a certain generalization of the determinant for an $n\times n$ square matrix $A$, that is
$$D_k(A) := \sum_{(j_1,j_2,...,j_n):\,\, |\{j_1,...,j_n\}|=k} \prod _{r=1}^n A_{r,j_r}.\qquad (1\leq k \leq n)$$
Here the sum condition means, that the $n$ indices $(j_1,...,j_n)$ consist of exactly $k$ distinct indices. That is, for $k=n$ the index combinations $(j_1,...,j_n)$ are permutations as in the usual determinant, although their signum is not included, so $D_n(A)$ is the permanent of $A$.
For $k<n$ the index combinations $(j_1,...,j_n)$ are not permutations anymore. Visually, the products include all "paths" on the matrix, which visit each of the $n$ rows exactly once, but only cover $k$ columns.
For $k=1$ we finally have
$$D_1(A) = \sum_{k=1}^n \prod_{r=1}^n A_{r,k}$$
Does anyone know a name for this generalization $D_k$ of the permanent? Also hints on their effective computation are welcome :)