An integer partition $\lambda$ of $n$ can be represented as a tuple $\lambda = (a_1, \cdots, a_n)$ in $\mathbb{Z}^n$ of $n$ nonnegative numbers $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ such that $\sum_{i=1}^n a_i = n$. Note that I use exactly $n$ numbers and allow zeroes in the entries of $\lambda$.
Now take all partitions $\lambda$ of $n$, consider them as points in $\mathbb{Z}^n$ and take their arithmetic mean $\lambda_\text{avg}$ in $\mathbb{R}^n$. How should $\lambda_\text{avg} = (b_1, \cdots, b_n)$ look like?
I guessed that it may look like the limiting shape of a random partition as calculated by Vershik (as explained in here). The heuristic calculation tells that, after normalizing the shape by $\sqrt{n}$ so that we plot the points $(i / \sqrt{n}, b_i / \sqrt{n})$ for $1 \leq i \leq n$, then the points should be near the curve $C$ described by $e^{-\pi x / \sqrt{6}}+e^{-\pi y / \sqrt{6}} = 1$. Below, you can compare $\lambda_\text{avg}$ (normalized) and the curve $C$. While they seem to loosely match, they don't seem to match completely.
Are there any results of this kind, or are there any possible suggestions to literature for which I can investigate what is happening further?
