A000410 Number of singular n X n rational (0,1)-matrices.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more

Comments

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

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

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. Metropolis and P. R. Stein, On a class of (0,1) matrices with vanishing determinants, J. Combin. Theory, 3 (1967), 191-198.
• Miodrag Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
• Miodrag Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346.
• Index entries for sequences related to binary matrices

Crossrefs

Cf. A000409, A046747, A064230, A064231.

Formula

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

Extensions

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

a(8) from Vladeta Jovovic, Mar 28 2006