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

A 1-factor of a hypergraph G=(X,W) is a set of hyperedges such that every vertex of G is incident to exactly one hyperedge from the set. A 1-factorization is a partition of all hyperedges of G into disjoint 1-factors. The adjacency matrix of a d-uniform hypergraph G is the d-dimensional (0,1)-matrix of order |X| such that an element aα1,…,αd of A equals 1 if and only if α1,…,αd is a hyperedge of G. Here 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.