I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague.
Let $(x_{ij})$ be an $n \times n$ matrix, and define $p(T_1,\ldots,T_n) = \det (x_{ij} - \delta_{ij}T_i)$. Let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $(x_{ij})$ counted with algebraic multiplicity. The problem is to find a "reasonable" general formula for $$ \prod_{\sigma \in \mathfrak{S}_n} p(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(n)}).$$
For example, when $n=1$ this expression vanishes identically. When $n=2$ one gets $x_{12}x_{21}\Delta$, where $\Delta = (\lambda_1-\lambda_2)^2$ is the discriminant of the characteristic polynomial. A conceptual way of seeing that this is the right answer for $n=2$ is that the formula must vanish on all lower or upper triangular matrices, and when the eigenvalues coincide all factors of the product vanish.
