Math 480, Spring 2014, Assignment 4

Following up on the discussion of symmetric polynomials, I was interested in counting the total number of such polynomials in n variables. Let \(\sigma_{k,n}\) be the number of symmetric polynomials in n variables where each monomial is of degree k. Let's look at \(\sigma_{4,4}\) and just list one representative monomial (lex order) for simplicity's sake.

\(wxyz, w^2xy, w^2x^2, w^3x, w^4\)

It should be clear that \(\sigma_{k,n} = P(k)\) where \(P(k)\) is the number of ways of partitioning k into addends, where order isn't considered. Starting with \(n=1\), the first ten terms are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42 and is listed in the Online Encyclopedia of Integer Sequences (OEIS) as [1] (which begins with \(n=0\).