The k-particle irreducible Brillouin conditions IBCk and the k-particle irreducible contracted Schrodinger equations ICSEk for a closed-shell state are analyzed in terms of a Moller-Plesset-type perturbation expansion. The zeroth order is Hartree-Fock. From the IBC2(1), i.e., from the two-particle IBC to first order in the perturbation parameter mu, one gets the leading correction lambda(2)((1)) to the two-particle cumulant lambda(2) correctly. However, in order to construct the second-order energy E-2, one also needs the second-order diagonal correction gamma(D)((2)) to the one-particle density matrix gamma. This can be obtained: (i) from the idempotency of the n-particle density matrix, i.e., essentially from the requirement of n-representability; (ii) from the ICSE1(2); or (iii) by means of perturbation theory via a unitary transformation in Fock space. Method (ii) is very unsatisfactory, because one must first solve the ICSE3(2) to get lambda(3)((2)), which is needed in the ICSE2(2) to get lambda(2)((2)), which, in turn, is needed in the ICSE1(2) to get gamma((2)). Generally the (k+1)-particle approximation is needed to obtain E-k correctly. One gains something, if one replaces the standard hierarchy, in which one solves the ICSEk, ignoring lambda(k+1) and lambda(k+2), by a renormalized hierarchy, in which only lambda(k+2) is ignored, and lambda(k+1) is expressed in terms of the lambda(p) of lower particle rank via the partial trace relation for lambda(k+2). Then the k-particle approximation is needed to obtain E-k correctly. This is still poorer than coupled-cluster theory, where the k-particle approximation yields Ek+1. We also study the possibility to use some simple necessary n-representability conditions, based on the non-negativity of gamma((2)) and two related matrices, in order to get estimates for gamma(D)((2)) in terms of lambda(2)((1)). In general these estimates are rather weak, but they can become close to the best possible bounds in special situations characterized by a very sparse structure of lambda(2) in terms of a localized representation. The perturbative analysis does not encourage the use of a k-particle hierarchy based on the ICSEk (or on their reducible counterparts, the CSEk), it rather favors the approach in terms of the unitary transformation, where the k-particle approximation yields the energy correct up to E2k-1. The problems that arise are related to the unavoidable appearance of exclusion-principle violating cumulants. The good experience with perturbation theory in terms of a unitary transformation suggests that one should abandon a linearly convergent iteration scheme based on the ICSEk hierarchy, in favor of a quadratically convergent one based on successive unitary transformations. (C) 2004 American Institute of Physics.