On the numbers of 1-factors and 1-factorizations of hypergraphs

A hypergraph G = (X,W ) is called d-uniform if each hyperedge w is a set of d vertices. A 1-factor of a hypergraph G is a set of hyperedges such that every vertex of the hypergraph is incident to exactly one hyperedge from the set. A 1factorization of G is a partition of all hyperedges of the hypergraph into disjoint 1-factors. The adjacency matrix of a d-uniform hypergraph G is the d-dimensional (0,1)matrix of order |X| describing sets of vertices of G such that they make a hyperedge. We estimate the number of 1-factors of uniform hypergraphs and the number of 1-factorizations of complete uniform hypergraphs by means of permanents of their adjacency matrices.