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: $f\_{31}=f\_3\*f\_1-f\_4$. 



Generalized Newton Identities
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](http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/ "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](https://math.stackexchange.com/questions/395842/decomposition-of-products-of-monomial-symmetric-polynomials-into-sums-of-them).


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. 


