The main propose is to develop the tensor permanent and the tensor combinatorial determinant, and to consider more interesting results for nonnegative tensors. There exist two forms for generalizing the determinant from matrices to tensors: the combi-natorial determinant and the geometrical determinant (sometimes called as hyperdeter-minant). We see that the tensor permanent and the tensor combinatorial determinant are two special cases of the tensor immanant. We derive some properties for the tensor permanent and consider the sign nonsingular tensors based on the combinatorial deter-minant. We generalize results from doubly stochastic matrices to totally plane stochastic tensors and obtain a probabilistic algorithm for locating a positive diagonal in a nonneg-ative tensor under certain conditions. Finally, we investigate how to give a lower bound for the minimum of the axial m-index assignment problem by means of plane stochastic tensors.