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Number of singular n X n rational (0,1)-matrices.
(Formerly M4308 N1803)
10

%I M4308 N1803 #28 Jun 05 2020 15:05:43

%S 0,0,6,425,65625,27894671,35716401889,144866174953833

%N Number of singular n X n rational (0,1)-matrices.

%C Number of all n X n (0,1)-matrices having distinct, nonzero ordered rows and determinant 0 - compare A000409.

%C a(n) is the number of singular n X n rational {0,1}-matrices with no zero rows and with all rows distinct, up to permutation of rows and so a(n) = binomial(2^n-1,n) - A088389(n). Cf. A116506, A116507, A116527, A116532. - _Vladeta Jovovic_, Apr 03 2006

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H N. Metropolis and P. R. Stein, <a href="https://doi.org/10.1016/S0021-9800(67)80006-1">On a class of (0,1) matrices with vanishing determinants</a>, J. Combin. Theory, 3 (1967), 191-198.

%H Miodrag Zivkovic, <a href="https://arxiv.org/abs/math/0511636">Classification of small (0,1) matrices</a>, arXiv:math/0511636 [math.CO], 2005.

%H Miodrag Zivkovic, <a href="https://doi.org/10.1016/j.laa.2005.10.010">Classification of small (0,1) matrices</a>, Linear Algebra and its Applications, 414 (2006), 310-346.

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%F n! * a(n) = A046747(n) - 2^(n^2) + n! * binomial(2^n -1, n).

%Y Cf. A000409, A046747, A064230, A064231.

%K nonn,nice,more

%O 1,3

%A _N. J. A. Sloane_

%E n=7 term from Guenter M. Ziegler (ziegler(AT)math.TU-Berlin.DE)

%E a(8) from _Vladeta Jovovic_, Mar 28 2006