It is stated that any symmetric function can be expressed in terms of the elementary symmetric polynomials. I am trying to do that for the following generating function:
\begin{equation} \prod_{1 \leq i<j \leq k} (x_{i} + x_{j}). \end{equation}
I can't quite seem to find a way to do this in general. I can see some kind of pattern though, involving the monomial symmetric functions, which are
\begin{equation} m_{(1,2)} = x_1 x_2^2 + x_1^2 x_2, \end{equation} and similarly for any $k$. The subscripted values in $m$ can be any integer.
I've shown that this can be done for the cases $k=1,2$ in which cases we get
\begin{equation} \prod_{1 \leq i<j \leq 2} (x_{i} + x_{j}) = (x_1 + x_2) = \frac{m_{(1,2)}}{\sigma_2^{(2)}}, \end{equation} \begin{equation} \prod_{1 \leq i<j \leq 3} (x_{i} + x_{j}) = \frac{m_{(1,2,3)}}{\sigma_3^{(3)}} + 2 \sigma_3^{(3)}, \end{equation} and for $k=4$ I get something in the form of \begin{equation} \prod_{1 \leq i<j \leq 4} (x_{i} + x_{j}) = \frac{m_{(1,2,3,4)}}{\sigma_4^{(4)}} + ?, \end{equation} which I cannot work out, at least not easily, though I will do the tedious calculation. (Is there a way to employ the monomial symmetric functions in mathematica somehow?)
Main question, is there already a known decomposition to this generating function? If so, what is it? If not, how should I go about making one? Or at least being able to determine it for values up to $k=6$.
So it turns out a very similar question has been answered quite nicely here: https://math.stackexchange.com/a/2403583/653042
I guess I should either delete my original question now, or make it more focused on the monomial symmetric functions instead...?