Partially decomposable and totally indecomposable nonnegative matrices

Abstract

ABSTRACT. We consider m x n ( m _~ n) matrices with entries from an arbitrary given finite set of nonnegative real numbers, including zero. In particular, (O, 1)-matrices are studied. On the basis of the classification of s-ch matrices by type and of the general formula for the number of matrices of nullity t valid for t > n and t _~ n > m (see [2]), an asymptotic (as n ~ oo) expansion is obtained for the total number of: (a) totally indecomposable matrices (Theorems 1 and 5)1 (b) partially decomposable matrices of given nullity t ~_ n (Theorems 2 and 4), (c) matrices with zero permanent (without using the inclusion-exclusion principle; Corollary of Theorem 2).We consider m x n (m < n) nonnegative matrices with entries from a given set R + of q nonnegative real numbers, including zero. If such a matrix A has zero submatrices of dimensions r x s and r + s = t, then the greatest of these numbers t is called the nullity of the matrix A (see [1]).In w we obtain an asymptotic (as n ---, r162 expansion, valid for t > n, for the number of matrices of nullity t and also for the number of semi-totally indecomposable matrices (m < n, t < n), which are important in the study of matrices with zero permanent (see [2]). For the number f(m, n) of matrices with zero permanent, an asymptotic expansion is obtained without using the inclusion-exclusion principle and Bonferroni's inequalities (as in [2]). In w an asymptotics for the number of totally indecomposable matrices of order n is derived. It should be noted that our investigation includes the case of (0, 1)-matrices (for q = 2), and we thus establish the asymptotics for the number of partially decomposable and totaUy indecomposable m x n (m < n) (0, 1)-matrices as n ~ co. w Partially decomposable matricesLet R~ be a fixed set of q nonnegative numbers, including zero. We consider the set M(m, n) of all possible m x n matrices, m <_ n, with elements from R~. A nonnegative matrix A E M(m, n) is said to be partially decomposable if its nullity is greater than or equal n. A matrix that is not partially decomposable is called semi-totally indecomposable (see [1]), and in the case m = n it is called totally indecomposable. We apply these definitions to a matrix A having dimensions m x n, with m > n, if they are satisfied by its transpose A T . Let f(m, n), g(m, n), and h(m, n), respectively, denote the number of matrices from M(m, n) with zero permanent, the number of matrices with nonzero permanent, and the number of those from M(m, n) that are semi-totally indecomposable.' We also assume that g(0, n) = h(0, n) -1 for any natural n. Let lPIt(rn, n) denote the set of all matrices of nullity t, and let ct(m, n) be their number. Next, we introduce the notion of type for matrices from Mr(m, n); namely, a matrix A E M't(m, n) is of type s if it contains a zero submatrix Ors of dimensions r x s, r + s = t, and for any of its zero submatrices O,,,, where u + v = t, the relation v >_ s is valid. Let M}S)(m, n) denote the set of matrices of nullity t and type s, and let clS)(rn, n) be t...