A000410 Number of singular n X n rational (0,1)-matrices. (Formerly M4308 N1803)

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

Offset

1,3

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

Table of n, a(n) for n=1..8.

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

Formula

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

Crossrefs

Cf. A000409, A046747, A064230, A064231.

Sequence in context: A162088 A199253 A199198 * A370961 A275686 A173760

Adjacent sequences: A000407 A000408 A000409 * A000411 A000412 A000413

Keyword

nonn,nice,more

Author

N. J. A. Sloane

Extensions

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

a(8) from Vladeta Jovovic, Mar 28 2006

Status

approved