Two alternative conditions for the stationarity of the energy expectation value with respect to k-particle excitations are the k-particle Brillouin conditions BCk and the k-particle contracted Schrodinger equations, CSEk. These conditions express the k-particle density matrices gamma (k) in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the gamma (k) in terms of their cumulants lambda (k), but this is not sufficient to make the BCk or CSEk separable (extensive), i.e., they are not expressible in terms of only connected diagrams. However, in a formulation based on the recently introduced general normal ordering with respect to arbitrary wave functions, the irreducible counterparts IBCk and ICSEk of the BCk and CSEk can be defined. They are easily evaluated explicitly in terms of the generalized Wick theorem for arbitrary wave functions, and they lead to equations for the direct construction of the cumulants lambda (k), which are additively separable quantities and which scale linearly with the system size. The IBCk or the ICSEk are necessary conditions for gamma and the lambda (k) to represent an exact n-fermionic eigenstate of the given Hamiltonian. To specify the desired state, additional conditions must be satisfied as well, e.g., the partial trace relations which relate lambda (2) to gamma and gamma (2). The particle number and the total spin must be specified and n-representability conditions enter implicitly. While the nondiagonal elements of gamma and the lambda (k) are determined by the IBCk or the ICSEk, the additional conditions mainly serve to fix the diagonal elements. A hierarchy of k-particle approximations is defined. It is based on the fact that the expansion in terms of cumulants lambda (k) can be truncated at any particle rank, which would not be possible for the density matrices gamma (k). For closed-shell states the one-particle approximation agrees with Hartree-Fock. (C) 2001 American Institute of Physics.