Let $t_1,\dots,t_n$ be indeterminates and consider the polynomial ring $\Bbb Z[t_1,\dots,t_n]$. Define two monomials in $t_1,\dots,t_n$ to be equivalent if some permutation of $t_1,\dots,t_n$ transforms one into the other. Define $\sum t_1^{a_1}\cdots t_r^{a_r}$ to be the summation of all monomials in $t_1,\dots,t_n$ which are equivalent to $t_1^{a_1}\cdots t_r^{a_r}$. For example, $\sigma_k=\sum t_1t_2\cdots t_k$, where $\omega_k$ is the $k$-th elementary symmetric polynomial.
Now let $S^k$ be the group of all homogeneous symmetric polynomials of degree $k$ in $t_1,\dots,t_n$. How can we show that the polynomials $\sum t_1^{a_1}\cdots t_r^{a_r}$, where $a_1,\dots,a_r$ ranges over all partitions of $k$ with length $r\leq n$, form a basis of $S^k$?