Partially decomposable and totally indecomposable nonnegative matrices

We consider m x n (m less than or equal to n) matrices with entries from an arbitrary given finite set of nonnegative real numbers, including zero. In particular, (0, 1)-matrices are studied. On the basis of the classification of such matrices by type and of the general formula for the number of matrices of nullity t valid for t > n and t greater than or equal to n > m (see [2]), an asymptotic (as n --> infinity) expansion is obtained for the total number of: (a) totally indecomposable matrices (Theorems 1 and 5), (b) partially decomposable matrices of given nullity t greater than or equal to n (Theorems 2 and 4), (c) matrices with zero permanent (without using the inclusion-exclusion principle; Corollary of Theorem 2).