Inequality with a Shatten norm

Question

Let $E_{\pi}$ denotes an expectation with respect to the normalized counting measure on $\Pi_n$, the group of all permutations of the set $\{1, \ldots, n\}$. Let $A_i, i=1, \ldots, n$ be a positive semi-definite $N\times N$ matrices. The Shatten $p$-norm is defined as $$ \|A\|_{S{p}}=\|\sigma(A)\|_p, 1\leq p <\infty. $$ Show that $$ E_{\pi}\|\sum_iA_{\pi(i)}\|_{S_p}\leq \|\sum_i|A_i|\|_1, $$ where $\|\cdot\|_1$ is a trace class norm.