where each product term represents one factorization symbolically and implicitly includes a sum over all factorizations within the class of terms identified. The purely correlated part is denoted by Δ Δ ⟨ ⟨ N ⟩ ⟩ . From these, the two-particle correlations Δ Δ ⟨ ⟨ 2 ⟩ ⟩ determine doublets while the three-particle correlations Δ Δ ⟨ ⟨ 3 ⟩ ⟩ are called triplets.