 Three-point density correlation of an ideal quantum gas

For a number-conserving Gaussian pairing state, six permutations give this expression. Each transposition has sign $ϵ$, positive for bosons and negative for fermions; the two three-cycles have positive sign for either statistics. Equivalently it is the permanent or determinant of the three-by-three one-body density matrix. For bosonic number states, repeated-mode factorial corrections must be included instead of assuming that Wick theorem.