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

0,5

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

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=0..33.

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

Formula

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

Example

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;
  ...

Prog

(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<n, ", ", "; "))); ) }

Crossrefs

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

Main diagonal gives A055165.

Sequence in context: A199082 A358644 A220669 * A286331 A363036 A354741

Adjacent sequences: A064227 A064228 A064229 * A064231 A064232 A064233

Keyword

nonn,nice,tabl

Author

N. J. A. Sloane, Sep 23 2001

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

Status

approved