A generalization of Waring's formula
`Waring's formula' expresses the power sum symmetric functions \(p_n\) as polynomials in the elementary symmetric functions \(e_k\) in the same variables. It states that \(p_n=n\sum(-1)^{n-l}(l-1)!\prod_i(e^{m_i}_i/m_i!)\), where the sum is over all partitions \(\lambda\) of \(n\), \(m_i\) is the number of parts of \(\lambda\) which are equal to \(i\), and \(l=l(\lambda)=\sum_i m_i\) is the length of \(\lambda\). The author gives a similar expression for the \(n\)th power sum of the products of the variables taken \(k\) at a time, i.e. the monomial symmetric function \(m_\mu\) corresponding to the partition \(\mu=(n^k)\). To state the result, we write \(u(\lambda)\) for the multinomial coefficient \(\begin{pmatrix} l\\ m_1,m_2,\dots\end{pmatrix}\), and \(e_\lambda\) for the product of the elementary symmetric functions corresponding to the parts of \(\lambda\). Then the coefficient \(A_\lambda\) of \((-1)^{k(n+1)}e_\lambda\) in \(m_\mu\) is obtained as a sum over all decompositions \(\lambda= \lambda_1\cup\lambda_2\cup\cdots\cup\lambda_r\) (meaning that the parts of \(\lambda\) are the parts of all the \(\lambda_i\) arranged in descending order) such that each \(\lambda_i\) is a partition of a multiple of \(n\). Each such decomposition contributes a summand \[ {(-1)^{l-r}\over \alpha_1!\alpha_2!\alpha_3!\dots} \prod^r_{i=1} {nu(\lambda_i)\over l(\lambda_i)} \] to \(A_\lambda\), where \(\alpha_1,\alpha_2,\alpha_3,\dots\) are the multiplicities with which different partitions appear in the decomposition. In the case \(k=1\), each partition \(\lambda\) admits only the trivial decomposition, and we recover Waring's formula in the form \(p_n= n\sum(-1)^{n-l}u(\lambda)l^{-1}e_\lambda\). In particular for \(\lambda=\mu'=(k^n)\) the coefficient of \(e^n_k\) in \(m_\mu\) is equal to 1. Comparison with the formula yields a special case of a classical identity of Sylvester.
scientific article; zbMATH DE number 934982