Computing the partition function for perfect matchings in a hypergraph

Given non-negative weights wS on the k-subsets S of a km-element set V, we consider the sum of the products wS₁ .. wS_m over all partitions V = S₁ .. S_m into pairwise disjoint k-subsets Si. When the weights wS are positive and within a constant factor of each other, fixed in advance, we present a simple polynomial-time algorithm to approximate the sum within a polynomial in m factor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman†"Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.