A064230 Triangle T(n,k) = number of rational (0,1) matrices of rank k (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 9, 6, 1, 49, 288, 174, 1, 225, 6750, 36000, 22560, 1, 961, 118800, 3159750, 17760600, 12514320, 1, 3969, 1807806, 190071000, 5295204600, 34395777360, 28836612000, 1, 16129, 25316928, 9271660734, 1001080231200, 32307576315840

Offset: 0

Author

N. J. A. Sloane, Sep 23 2001

Keywords

• nonn
• nice
• tabl

Comments

Rows add to 2^(n^2).

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

Table 3 from M. Zivkovic, Classification of small (0,1) matrices (see link). - Vladeta Jovovic, Mar 28 2006

Examples

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   9,      6;
  1,  49,    288,     174;
  1, 225,   6750,   36000,    22560;
  1, 961, 118800, 3159750, 17760600, 12514320;
  ...
		

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

• M. Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346.

Crossrefs

Cf. A064231, A000409, A000410, A046747, A086875.

Main diagonal gives A055165.

Programs

PARI

T=matrix(5,5); { for(n=0,4, mm=matrix(n,n); for(k=0,n,T[1+n,1+k]=0); forvec(x=vector(n*n,i,[0,1]), for(i=1,n, for(j=1,n,mm[i,j]=x[i+n*(j-1)])); T[1+n,1+matrank(mm)]++); for(k=0,n,print1(T[1+n,1+k], if(k
				

Formula

Sum_{k=1..n} k * T(n,k) = A086875(n). - Alois P. Heinz, Jun 18 2022

Extensions

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

6 more terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Dec 17 2004

More terms from Vladeta Jovovic, Mar 28 2006