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 $\mathrm{\mid }X\mathrm{\mid }$ such that an element $a_{\alpha_1, \ldots, \alpha_d}$of$A$equals1ifandonlyif$\left{\alpha_1, \ldots, \alpha_d\right}$isahyperedgeof$G$. 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.