  Given non-negative weights w_S on the k-subsets S of a km-element set V, we
consider the sum of the products w_{S_1} ... w_{S_m} for all partitions V = S_1
cup ... cup S_m into pairwise disjoint k-subsets S_i. When the weights w_S are
positive and within a constant factor, fixed in advance, of each other, 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.
