Suppose that we have $n\in \mathbb{Z}_{+}$ and some $\alpha\ge 3$.
I am trying to find maximum value of:
$\sum_{i,j=1}^{n}|\lambda_{i}-\lambda_{j}^{*}|^{\alpha},$
over
$\{\lambda\in \mathbb{Z}^{n}: n\ge \lambda_{1}\ge \cdots\ge \lambda_{n} \ge 0 \},$
where
$\lambda_{j}^{*}=|\{i: \lambda_{i}\ge j\}|$ - $\lambda^{*}$ is conjugate partition of $\lambda$.
I believe - due to some computer simulations - that argmax solution satisfies
a) $\lambda=\lambda^{*}$
b) $|\{j:\lambda_{j}\not=\lambda_{j+1}\}| \le 1$,
but I do not know how I could prove it. I will be glad for any ideas or insights.