Generalized Newton Identities I learnt a lot of new words (Hall-Littlewood, Jack and Macdonald polynomials) but unfortunately everything I dug up is written without a single example and I still don't know the answer to a very obvious generalization: Let $P$ be a partition. Example: $31$. The symmetric polynomial corresponding to $P$ would be $f_{31}=a^3*b+a^3*c+...$ (with any number of variables: if we work in the ring of symmetric functions, it doesn't matter anyway). Now Newton's identities cover $f_{1...1}$ vs. $f_n$. I'd like to have an explicit formula or at least algorithm (at the moment I do it by hand, which is tedious and error-prone) to express any $f_P$ in the basis functions (actually I need the power sums $f_n$, but since the standard Newton identities cover the conversion, a formulation via the elementary symmetric polynomials is fine, either). Example with power sums: $f_{31}=f_3*f_1-f_4$.        
 A: As far as I understand, what you call $f_P$ is usually called monomial symmetric function and is denoted $m_P$. So I interpret your question as asking for an algorithm converting monomial symmetric funtions to power sum symmetric functions. Such an algorithm seems to be described here: 
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4656270/
Also there is one which was implemented in Symmetrica (which is the one used in Sagemath) more than twenty years ago. I don't think the algorithm was documented anywhere but as a comment in the code (the file is called tmp.c) . The comment seems to indicate that they use a recursive divide and conquer method using the following recursion step:
$$
  m_{a_1,a_2,...,a_n,a_{n+1},...a_{2n}} =
   m_{a_1,...,a_n} * m_{a_{n+1},..,a_{2n}} - \text{terms of length} <2n
 $$
You can compute the rest of the terms using this answer.
Anyway, unless you really need to say something about the algorithm, I would recommend using a computer algebra system which already has this algorithm implemented. 
