Let $V_n$ be the standard permutation representation of the symmetric group $S_n$, and let $\mathbb{S}_{\lambda}$ denote the Schur functor associated to the partition $\lambda$.

Let $\lambda$ range over all partitions of some natural number $k$. We may ask for the smallest $n$ making the characters of the representations $\mathbb{S}_{\lambda} V_n$ linearly independent. A little experimentation indicates that $n=k$.

If instead $\lambda$ ranges over all partitions of natural numbers less than or equal to $k$, it seems that $n=2k$.

Are these patterns correct?