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Reference request on symmetric polynomials
Not a complete answer, but note that
\begin{align}
(*) \;\;\;e_k =e_k(x_1,\ldots,x_n)=\sum x_{i_1}x_{i_2}\cdots x_{i_k},
\end{align}
where the sum is over $1 \leq i_1 < i_2 < \ldots < i_k \leq n$. Mo …