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

Author

N. J. A. Sloane, Sep 23 2001

Keywords

• nonn
• nice
• tabl

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.

Examples

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

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.

Crossrefs

Cf. A064230, A000409, A000410, A046747.

Programs

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
				

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