Labelling the character tables of symmetric and alternating groups

Let $X$ be a character table of the symmetric group $S_n$. It is shown that unless $n = 4$ or $n=6$, there is a unique way to assign partitions of $n$ to the rows and columns of $X$ so that for all $\lambda$ and $\nu$, $X_{\lambda\nu}$ is equal to $\chi^\lambda(\nu)$, the value of the irreducible character of $S_n$ labelled by $\lambda$ on elements of cycle type $\nu$. Analogous results are proved for alternating groups, and for the Brauer character tables of symmetric and alternating groups.