  Let the formal power series f in d variables with coefficients in an
arbitrary field be a symmetric function decomposed as a series of Schur
functions, and let f be a rational function whose denominator is a product of
binomials of the form (1 - monomial). We use a classical combinatorial method
of Elliott of 1903 further developed in the Partition Analysis of MacMahon in
1916 to compute the generating function of the multiplicities (i.e., the
coefficients) of the Schur functions in the expression of f. It is a rational
function with denominator of a similar form as f. We apply the method to
several problems on symmetric algebras, as well as problems in classical
invariant theory, algebras with polynomial identities, and noncommutative
invariant theory.
