Let a Gaussian vector. Assuming that , how to do you prove that:

where is a partition of {1,...,2n} into n pairs, and the summation extends over all the partitions (the permutation of elements of a pari is considered to yied to the same partition)?

The natural approach is to prove it by induction using the usual results on (partial) derivatives of the characteristic function and the moments, assuming the Gaussian vector is represented using a correlation matrix. However, the calculation, albeit tedious, cannot be concluded without some results on permutation to carry on the induction hypothesis.

This has probably to do with decomposition of permutations into cycles, so I was wondering if anyone had any idea about how to proceed further.

Thanks in advance.