A086875 -id:A086875 - cp's OEIS frontend

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

N. J. A. Sloane, Sep 23 2001

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

			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;
  ...

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.

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

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

Main diagonal gives A055165.

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

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

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

A086098 Sum of rank(M) over all n X n matrices over GF(2).

Original entry on oeis.org

1, 21, 1141, 208965, 139889701, 354550756581, 3464730268306021, 131934922593867875685, 19707939574875773323508581, 11599530748705611712884878698341, 26983642577843418550426409405086580581, 248652621703069011230281370429818425958461285

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 24 2003

nonn

a(n) <= A086875(n).

Andrew Howroyd, Table of n, a(n) for n = 1..50
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A086190, A086207, A086875.

a(n) = {my(q=2); sum(r=1, n, r*prod(j=0, r-1, (q^n-q^j)^2/(q^r-q^j)))} \\ Andrew Howroyd, Jul 08 2018

For prime power q the number of rank-r n X n matrices over GF(q) is F(r, n) = product j=0..(r-1) (q^n-q^j)^2/(q^r-q^j) so a(n) = sum r=1..n r*product j=0..(r-1) (q^n-q^j)^2/(q^r-q^j) . In this case q=2.

a(n) = Sum_{r=1..n} r*Product_{j=0, r-1} (2^n - 2^j)^2/(2^r - 2^j). - Andrew Howroyd, Jul 08 2018

Terms a(8) and beyond from Andrew Howroyd, Jul 08 2018

cp's OEIS Frontend

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

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