nth power symmetric polynomial in terms of Schurs polynomial The Schur's polynomial forms the basis of the symmetric algebra so does the power symmetric function. nth power symmetric function are the function of the form $\sum_i x_i^n$. Let $\lambda \vdash n$ be an partition of $n$. Then we have the famous Frobenius relationship
$$
  s_{\lambda}=\frac{1}{n!}\sum_{\pi\in \mathbb{S_n}}\chi^{\lambda}(\pi)p_{\pi}. 
$$
So to get the $p_{\lambda}$ in terms of $s_{\lambda}$ we can just inverse the matrix obtained by the above equation by varying $\lambda$ over all the partition of $n$.
My question is that there is a nice expression for it and any combinatorial explanation ?
 A: Yes there is an easy formula. With respect to the usual inner product on what you call the symmetric algebra (really the algebra of symmetric functions), for which the Schur functions form an orthonormal basis, the monomials $p_\mu$ in the power sums form an orthogonal (but not orthonormal) basis. This means that the coefficient of $p_\mu$ when expressing some element $a$ of the algebra in this basis is given by $\frac{\langle a,p_\mu\rangle}{\langle p_\mu,p_\mu\rangle}$. For the case $a=s_\lambda$, the inner product $\langle s_\lambda,p_\mu\rangle$ equals the character value $\chi^\lambda(C_\mu)$ of the irreducible character $\chi^\lambda$ at any permutation of cycle type $\mu$. Also the inner product $\langle p_\mu,p_\mu\rangle$ can be interpreted as the order of the centraliser in$~S_n$ of such a permutation, so that $\frac1{\langle p_\mu,p_\mu\rangle}$ is equal to the fraction $\frac{\#C_\mu}{n!}$ of the elements of $~S_n$ that have cycle type$~\mu$; the Frobenius formula follows.
Now since by definition the Schur functions $s_\lambda$ form an orthonormal basis, the coefficient of $s_\lambda$ when expressing some element $a$ of the algebra in this basis is simply $\langle a,s_\lambda\rangle$. For $a=p_\mu$ we have seen that this is equal to the character value $\chi^\lambda(C_\mu)$, so one gets the simple formula
$$
  p_\mu=\sum_{\lambda\vdash n}\chi^\lambda(C_\mu)s_\lambda.
$$
The character values can be concretely computed using the Murnaghan-Nakayama rule.
