Geodesic
Partially decomposable and totally indecomposable nonnegative matrices
Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 643-662.

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We consider $m\times n$, $m\le n$, 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 $t\ge n>m$ (see [2]), an asymptotic (as $n\to\infty$) expansion is obtained for the total number of: (a) totally indecomposable matrices (Theorems 1 and 5), (b) partially decomposable matrices of given nullity $t\ge n$ (Theorems 2 and 4), (c) matrices with zero permanent (without using the inclusion-exclusion principle; Corollary of Theorem 2). 
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     author = {Yu. V. Bolotnikov},
     title = {Partially decomposable and totally indecomposable nonnegative matrices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {643--662},
     publisher = {mathdoc},
     volume = {59},
     number = {5},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a0/}
}
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Yu. V. Bolotnikov. Partially decomposable and totally indecomposable nonnegative matrices. Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 643-662. http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a0/

[1] Fennet T. I., Loizou G., “Combinatorial aspects of rectangular non-negative matrices”, Discrete Math., 20 (1977), 217–234 | MR

[2] Bolotnikov Yu. V., Tarakanov V. E., “Neotritsatelnye matritsy s nulevym permanentom”, Matem. zametki, 58:4 (1995), 493–504 | MR | Zbl

[3] Sachkov V. N., Veroyatnostnye metody v kombinatornom analize, Nauka, M., 1978 | Zbl