Nice update! $\det(I - xA)^{-1} = \det(I + xA + x^2 A^2 + \cdots)$. Then expanding this by multilinearity of the determinant, and taking the coefficient of $x^k$ one gets the right hand side of the general formula, and taking the Jordan form of $A$ and using the generating function of complete homogeneous symmetric polynomials one gets the left hand side. Power sum symmetric polynomials are easy too, of course, via the trace. The proof for the general case is not much more difficult than these special cases though :)

It can be generalized even beyond subsets. A subset of the plane may be viewed as a function $f : \mathbb{R}^2 \to {0,1}$, but we can also generalize the range to be $\mathbb{R}$, to allow fractional points to be in that subset. We can then define the integral over the "generalized subset" $\int_f g$ = $\int f g$. In this case Greene's theorem becomes the same as integration by parts. Now we can generalize $f$ further from a function to a Schwartz distribution, and perhaps beyond.

Thanks a lot for the screenshots!

Thank you! So in terms of the formula, we get $s_k$ on the left hand side if we take $p_i = 1$ if the tuple $i$ has $k$ ones and $n-k$ zeroes, and $p_i = 0$ otherwise. Then some of the $A$'s on the right hand side get raised to the 1st power and some to the 0th power, and after Laplace expansion on the columns that have the 0th power, we get a sum over all $k\times k$ principal minors of $A$. I'll try to get access to those books. Is by any chance the proof by looking at the coefficient of $x^k$ in $\det(A + xI)$? I believe I have seen that proof before. P.S. I loved your lectures at Leiden!

I put a concise version of my proof in the question.

Ah, now I understand. I misunderstood your initial comment and thought there was a particular known family of functions $H$ that you were talking about, but you are talking about the $p_i$ that are of that form for any $H$. Sorry for the confusion and thank you for the explanation.

Thanks! I think I understand all the steps except one. I could not understand why the linear span is the same as the algebra. The degree of these special $p_i$ seems to be $\leq n$ so I'd expect that the same holds true for their span, whereas in general the $p_i$ can have any degree. Could you help me understand where my reasoning went wrong? By the way, it is easy to prove the reverse implication, that any symmetric polynomial can be written in elementary symmetric polynomials, by taking $A$ to be the companion matrix of a polynomial in the identity :)