 Ordered Grassmann Gaussian contractions

Use the Berezin integral orientation $\int {\prod }_{i}d{\bar{\theta }}_{i}d{\theta }_i{\theta }_1{\bar{\theta }}_1\cdots {\theta }_n{\bar{\theta }}_n=1$. For an invertible matrix $B$, the normalized weight $e^{-\bar{\theta }B\theta }$ has $⟨{\theta }_i{\bar{\theta }}_j⟩=(B^{-1}{)}_{ij}$. Differentiate its determinant with respect to $B_{ji}$ to obtain this formula; two derivatives give the displayed four-point formula. The second pairing carries a fermionic sign. These are ordered contractions, so exchanging two odd factors changes their sign.