Let $A_k$ be a set of polinomials $f(x_1,x_2,\ldots,x_n)$ in $\mathbb{Q}[x_1,x_2,\ldots,x_n]$ symmetric on $k<n$ variables: $$f(x_{\sigma(1)},\ldots, x_{\sigma(k)},x_{k+1}, \ldots,x_n)=f(x_1,x_2,\ldots,x_n),\,\, \sigma \in S_k$$
$A_k$ is a $\mathbb{Q}-$ vector space. What is the basis or dimension of $A_k$?
The dimension is $\infty$. Note e.g. that $\sum_{j=1}^k x_j^p$ is in $A_k$ for all positive integers $p$.
But perhaps you meant to consider polynomials of fixed degree?