I am currently writing a function in Macaulay2 which given an elementary symmetric function $e_i$ outputs its expression in terms of power sums according the formula:
$$e_n=\sum_{|\lambda|=n}(-1)^{|\lambda|-l(\lambda)} z_{\lambda}^{-1}p_{\lambda}$$ where $|\lambda|$ is the size of the partition $\lambda$ and $l(\lambda)$ is its length, $p_{\lambda}=p_{\lambda_1}p_{\lambda_2}\cdots$ and $$z_{\lambda}=\prod_{i\geq 1}\left(i^{m_i}\cdot m_i!\right)$$ where $m_i$ is the number of parts of $\lambda$ equal to $i$.
I am close but due to my ignorance of the language I am missing two important parts of the program:
(1) A way to find all positive integer solutions to the equation $\sum_{i=1}^nix_i = m$ (i.e. the partitions of $m$ as $n$-tuples).
(2) Given a list of polynomials, a way to compute the product of all elements in that list.
Any help would be greatly appreciated.