Let A be a non-negative integer square matrix with eigenvalues x_{1}, x_{2}, ... x_{n}. Any symmetric function of these eigenvalues with integer matrices is an integer. I'm aware of the following results regarding the combinatorial interpretation of these integers:

If A is the adjacency matrix of a finite directed graph G, the power symmetric functions of the eigenvalues count closed walks on A with a distinguished starting point.

Similarly, the complete homogeneous symmetric functions of the eigenvalues count non-negative integer linear combinations of aperiodic closed walks on A.

(Gessel-Viennot-Lindstrom) If A

_{ij}is the number of paths from source i to sink j on, say, a 2-D lattice where the only permissible moves are to the right and up, then the elementary symmetric functions of the eigenvalues count the number non-intersecting k-tuples of paths from the sources to the sinks. In particular det A is the number of non-intersecting n-tuples of paths.

Do these results generalize to give a nice combinatorial interpretation of the value of the Schur function associated to an arbitrary partition evaluated at x_{1}, x_{2}, ... x_{n} in terms of some combinatorial object attached to A? What conditions need to be placed on A so that the Schur functions are always non-negative?

Feel free to either talk about the GL(n) perspective or to frame your discussion entirely in terms of tableaux.