Bases for symmetric polynomials I’ve been playing with symmetric polynomials, as one does, and I’ve run into something that must be familiar, but I can’t find anything about it. To present the idea, I’ll work with symmetric polynomials in $\Bbb Z[x,y]$, but the idea extends to any number of variables, with the details just getting a little more difficult to write down.
We have the elementary symmetric polynomials in two variables: $\sigma_1=x+y$ and $\sigma_2=xy$.
Now, we pick a degree, say $4$. If we want to express any degree $4$ symmetric polynomials in $x$ and $y$, there are two bases that seem natural to use:
$\rho_1=\sigma_1^4\\
\rho_2=\sigma_1^2\sigma_2\\
\rho_3=\sigma_2^2$
and
$\tau_1=x^4+y^4\\
\tau_2=x^3y+xy^3\\
\tau_3=x^2y^2$
These bases are related by the equation:
$\left[\begin{matrix} 1&4&6\\0&1&2\\0&0&1\end{matrix}\right]\left[\begin{matrix} \tau_1\\ \tau_2\\ \tau_3\end{matrix}\right] = \left[\begin{matrix} \rho_1\\ \rho_2\\ \rho_3\end{matrix}\right]$
The entries in this matrix are binomial coefficients, if we were working in 3 variables, we’d be looking at trinomial coefficients instead, etc.
My question is: what am I looking at? I assume this has been thoroughly studied. Are these bases, and the transformations between them, called something? Are the transformations special in some way? They all have characteristic polynomials of the form $(1-\lambda)^k$ for some $k$, being triangular with $1$s on the diagonal. Is there more that I’m not seeing? Is there a quicker way to write the transformations down than just expanding each $\rho_i$?
Thanks in advance for any insight or information anyone can provide.
 A: The change of basis matrix from the monomial basis (your $\{\tau_i\}$) to the elementary symmetric basis (your $\{\rho_i\}$) has a combinatorial description, which is given as Proposition 7.4.1 in Richard Stanley's Enumerative Combinatorics, Volume 2. I will give the statement here.
Recall that a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is a finite non-increasing list of positive integers. Associated with each partition is an elementary symmetric function $e_\lambda$ and a monomial symmetric function $m_\lambda$. For example, your basis elements are $\rho_1=e_{1111}$, $\rho_2=e_{211}$, $\rho_3=e_{22}$, $\tau_1=m_4$, $\tau_2=m_{31}$, and $\tau_3=m_{22}$. Both $\{e_\lambda\}$ and $\{m_\lambda\}$ are bases for symmetric functions. The change of basis has the form
$$
e_\lambda=\sum_\mu M_{\lambda\mu} m_{\mu}
$$
where for partitions $\lambda=(\lambda_1,\ldots,\lambda_k)$ and $\mu=(\mu_1,\ldots,\mu_\ell)$, the coefficient $M_{\lambda\mu}$ is the number of $k\times\ell$ matrices with entries in $\{0,1\}$ such that the sum of the $i$-th row is $\lambda_i$ and the sum of the $j$-th column is $\mu_j$.
