Is there an explicit formula expressing the [power sum symmetric polynomials](http://en.wikipedia.org/wiki/Power_sum_symmetric_polynomial) 
$$p\_k(x\_1,\ldots,x\_N)=\sum\nolimits\_{i=1}^N x\_i^k = x\_1^k+\cdots+x\_N^k$$ 
of degree $k$ in $N < k$ variables entirely through the power 
sum symmetric polynomials $p\_j(x\_1,\ldots,x\_N)$ of degrees $ j \le N $? 


Examples: 
$$N=1,\ k=2: \quad p\_2=x^2=x\times x=p\_1^2$$


$$N=2,\ k=3: \quad
p\_3 = x^3 + y^3 = [3(x^2+y^2)(x+y)-(x+y)^3]/2 = (3 p\_2 p\_1-p\_1^3)/2$$



> 
> What is the general formula?
> 
> 
> 


I am looking for a formula similar to that for the expansion of the [Schur functions](http://en.wikipedia.org/wiki/Schur_polynomial#Relation_to_representation_theory)
 $s\_\lambda$ in terms of the symmetric power sums:


$$ s\_\lambda=\sum\_{\rho=(1^{r\_1},2^{r\_2},3^{r\_3},\dots)}\chi^\lambda\_\rho \prod\_j \frac{p^{r\_j}\_j}{r\_j!},$$
where the coefficients $\chi^\lambda\_\rho$ are the characters of the representation of the symmetric group indexed by the partition $\lambda$ evaluated at elements of cycle type indexed by the partition $\rho=(1^{r\_1},2^{r\_2},3^{r\_3},\dots)$, which contains 
$ r\_j $ 
parts of length $j$. 


Clearly, the power sums of degree higher than $N$ can be expanded in a similar manner:
$$
p\_k=\sum\_{\rho}a\_{k;\rho}\prod\_{j=1}^N p\_j^{r\_j},
$$
where $\rho=(1^{r\_1},2^{r\_2},\dots,N^{r\_N})$ is the partition of $k$ such that
$k=r\_1+2r\_2+3r\_3+...+Nr\_N$. 


In the above example for $N=2,\ k=3$ one has $a\_{3;\ (1^{1},2^{1}) }=3/2$ and 
$a\_{3;\ (1^{3},2^{0})}=-1/2$. 



> 
> My question can be thus reformulated as follows:
>  given $r\_1,...,r\_N$ what is the
>  explicit formula for $a\_{k;\rho}$?
> 
> 
> 




---


**Note Added**


Actually, [Wikipedia](http://en.wikipedia.org/wiki/Newton%27s_identities#Expressions_as_determinants) tells us how to construct a certain explicit formula for $p\_k$.
It gives the following expressions for $p\_n$ with $n=N$ in terms of $ e\_j, $


$$
p\_n =
\begin{vmatrix}
e\_1 & 1 & 0 & \cdots & \\\ 
2e\_2 & e\_1 & 1 & 0 & \cdots & \\\
3e\_3 & e\_2 & e\_1 & 1 & \cdots & \\\
\vdots &&& \ddots & \ddots & \\\
ne\_n & e\_{n-1} & \cdots & & e\_1 &
\end{vmatrix},
$$


and for $e\_n$ with $n=N$ in terms of $ p\_j, $


$$
e\_n=\frac1{n!}
\begin{vmatrix}p\_1 & 1 & 0 & \cdots\\\ p\_2 & p\_1 & 2 & 0 & \cdots \\\ \vdots&& \ddots & \ddots \\\ p\_{n-1} & p\_{n-2} & \cdots & p\_1 & n-1 \\\ p\_n & p\_{n-1} & \cdots & p\_2 & p\_1
\end{vmatrix}.
$$


As far as I can see from the derivation described in Wikipedia, these determinant expressions are also valid for $p\_n$ with $ n > N $ and for $e\_n$ with $ n < N $.


For $p\_n$ with $n>N$ one should take into account that all $ e\_k=0 $ for $ k > N $, so that the resulting matrix has zeros in both right-upper and left-lower corners. 


Substituting the determinants for $e\_j$ into the determinant for $p\_k$, one gets the 
explicit formula which seems to solve the problem. 


However, I still don't know how to obtain the coefficients $a\_{k;\rho}$ in the expansion of $ p\_k $ in terms of the first $N$ power sums which would be the desired (really explicit) formula. 



Expressing power sum symmetric polynomials in terms of lower degree power sums
Assuming you have $n$ variables then for $k\geq n$, Robin Chapman's identity above can be written as
$$(p\_n,p\_{n-1},\dots, p\_1)\begin{pmatrix}
 e\_1 & 1 & \cdots & 0 \\\
 -e\_2 & 0 & \ddots & \vdots \\\
 \vdots & \vdots & \ddots & 1 \\\
 (-1)^{n-1}e\_n & 0 & \cdots & 0
 \end{pmatrix}^{k-n}=(p\_k,p\_{k-1},\dots, p\_{k-n+1})$$


Now to finish the job you need to express the $e\_i$'s in terms of the power sum symmetric functions too. This is given by $$e\_n=\sum\_{|\lambda|=n}(-1)^{|\lambda|-l(\lambda)} z\_{\lambda}^{-1}p\_{\lambda}$$ 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 thought I'd remark that the formulas you are quoting are all valid in $\Lambda\_{\mathbb{Q}}$, the ring of symmetric functions in *infinitely* many variables while the one you are searching for is not, because the $p\_\lambda$'s are an orthogonal basis in this ring with $\langle p\_{\lambda},p\_{\mu}\rangle =\delta\_{\lambda \mu}z\_{\lambda}$. This is also the same reason why the formula for Schur polynomials may contain arbitrary $p\_{\lambda}$'s in it. In fact the reason why that formula is important is because it gives you the transition from the basis of Schur polynomials to that of power sum polynomials in $\Lambda\_{\mathbb{Q}}$.


