For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S\_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$). I need a bound for $S(S\_k \times S\_{n-k}, \mathbb{Q})$ the best as possible. 



representations of $S_k \times S_j$
I guess that $S\_n$ denotes the symmetric group. It is well known that all irreps of $S\_n$ over $\mathbb C$ are defined over $\mathbb Q$. Therefore $S(S\_k\times S\_{n-k},\mathbb{C})=S(S\_k\times S\_{n-k},\mathbb{Q})$. Also the fact that the sum of the squares of the dimensions is the group order and Cauchy-Schwarz immediately imply that $S(G,\mathbb C)\le \sqrt{c(G)|G|}$ where $c(G)$ is the number the conjugacy classes. In your case it means $S(S\_n,\mathbb C)\le \sqrt{p(n)n!}$ where $p$ is the partition function.


