distribution of Young diagrams

Consider $\Lambda^p(C^n\otimes C^n)=\oplus_{\pi}S_{\pi}C^n\otimes S_{\pi'}C^n$ as a $GL_n\times GL_n$-module. This space has dimension $\binom {n^2}p$. I would like any information on the shapes of pairs of Young diagrams $(\pi,\pi')$ that give the largest contribution to the dimension asymptotically. I am most interested in the case where $p$ is near $n^2/2$. Is there a slowly growing function $f(n)$ such that partitions with fewer than $f(n)$ steps contribute negligibly? If so, can the fastest growing such $f$ be determined?

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What is denoted by $S_{\pi}C^n$? –  Fedor Petrov Feb 28 at 19:18
The Schur functor corresponding to $\pi$ applied to $\mathbb{C}^n$, I think. –  alpoge Feb 28 at 20:30