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I think one can at least give a First Fundamental Theorem in the spirit of really classical invariant theory. For the group $S_n$, I think you can build all invariants of vectors $(X_i)_{1\le i\le n}$ …

The notation and framing of the problem are not optimal, because the $n_i$'s being integers is a bit of a distraction. Let $[k]:=\{1,\ldots,k\}$, and let ${\rm Part}_k$ be the set of set partitions of …

For a polynomial or a formal power series $F(x_1,\ldots,x_N)$ in $x_1,\ldots,x_N$, let $$ {\rm Sym} [F(x_1,\ldots,x_N)]=\frac{1}{N!}\sum_{\sigma\in \mathfrak{S}_N} F(x_{\sigma(1)},\ldots,x_{\sigma(N) …

Another systematic study of multisymmetric polynomials is in the thesis of Emmanuel Briand available in .ps format at: http://emmanuel.jean.briand.free.fr/publications/polms/ see also his other public …

Not yet an answer but you can see $f$ as an element of the symmetric power $S^d(V^{\vee})$ for $V$ a vector space of dimension $n+1$. You can take $V=S^n(W)$ with $W$ of dimension 2. That should give …