How to prove the following identity:
$$\det(A)=\frac{1}{d!}\sum_{\sigma\in S_d}\mathrm{sgn}(\sigma)\mathrm{Tr}_{\sigma}(A)$$
where $\mathrm{Tr}_{\sigma}(A)$ is defined as following
if $\sigma$ is of type $1^{c_1}2^{c_2}\dots d^{c_d}$, then $\mathrm{Tr}_{\sigma}(A)=\prod_{i=1}^d(\mathrm{Tr}A^i)^{c_i}$