Bases for symmetric polynomials

Solution 1:

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 $\mu_j$.