A064231 Triangle read by rows: T(n,k) = number of rational (+1,-1) matrices of rank k (n >= 1, 1 <= k <= n).

2, 8, 8, 32, 288, 192, 128, 6272, 36864, 22272, 512, 115200, 3456000, 18432000, 11550720, 2048, 1968128, 243302400, 6471168000, 36373708800, 25629327360, 8192, 32514048, 14809546752, 1557061632000, 43378316083200, 281770208133120

Offset

1,1

Comments

Rows add to 2^(n^2) = A002416. - Jonathan Vos Post, Feb 27 2011

Komlos and later Kahn, Komlos and Szemeredi show that almost all such matrices are invertible.

References

J. Kahn, J. Komlos and E. Szemeredi: On the probability that a random +-1 matrix is singular, J. AMS 8 (1995), 223-240.

J. Komlos, On the determinants of random matrices, Studia Sci. Math. Hungar., 3 (1968), 387-399.

Links

Table of n, a(n) for n=1..27.

Example

2; 8,8; 32,288,192; 128,6272,36864,22272; ...

Prog

(PARI) T=matrix(4, 4); { for(n=1, 4, mm=matrix(n, n); for(k=1, n, T[n, k]=0); forvec(x=vector(n*n, i, [0, 1]), for(i=1, n, for(j=1, n, mm[i, j]=(-1)^x[i+n*(j-1)])); T[n, matrank(mm)]++); for(k=1, n, print1(T[n, k], if(k<n, ", ", "; "))); ) }

Crossrefs

Cf. A064230, A000409, A000410, A046747.

Sequence in context: A230739 A227326 A323852 * A181130 A212196 A156052

Adjacent sequences: A064228 A064229 A064230 * A064232 A064233 A064234

Keyword

nonn,nice,tabl

Author

N. J. A. Sloane, Sep 23 2001

Extensions

More terms and PARI code from Michael Somos, Sep 25 2001

More terms from Vladeta Jovovic, Apr 02 2006

Offset changed to 1 by T. D. Noe, Mar 02 2011

Status

approved