Assume we are given are fermionic many-particle system with creation operators $c_1^\dagger,c_2^\dagger,...$ acting on some Hilbert space $\mathcal{H}$. That is, the $c_i^\dagger$ and their corresponding annihilation operators $c_i=(c_i^\dagger)^\dagger$ act as bounded operators on $\mathcal{H}$ and satisfy the CAR relations $$\{c_i,c_j\}=\{c^\dagger_i,c_j^\dagger\}=0\quad\text{and}\quad\{c_i^\dagger,c_j\}=\delta_{ij}\quad\forall i,j$$

In the *bosonic* case, the Wick theorem can be elegantly written in the form
$$\prod_{j=1}^nc_{i_j}^{\sigma_j}=\sum_{Q\subset\mathbb{N}_n}\langle\prod_{j\in Q}^nc_{i_j}^{\sigma_j}\rangle:\prod_{j\not\in Q}^nc_{i_j}^{\sigma_j}:$$
for any $i_1,...,i_n\in\mathbb{N}$ and $\sigma_1,...,\sigma_n\in\{-1,1\}$, where $c^-_i:=c_i$, $c^+_i:=c^\dagger_i$ for any $i\in\mathbb{N}$, $\mathbb{N}_n:=\{1,\dots,n\}$, $\langle A\rangle:=\langle\Omega\mid A\Omega\rangle$ tenotes the vacuum expectation value and $:\cdots:$ denotes the Normal ordering. However, for *fermions* this formula has to be modified by attaching an appropriate sign to every of the summands. What is this sign?