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 $k=1$, 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 $k=0$). Then the total number of symmetric polynomials in n variables $= \sum_{k=1}^{n}P(k)$, which is the partial sum of the partition numbers. This new sequence (or series of prior sequence) begins with 1, 3, 6, 11, 18, 29, 44, 66, 96, 138 and is listed in the OEIS as [2] "number of sums S of positive integers satisfying S <= n". None of the comments thus far say "number of symmetric polynomials of degree n"