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 A064230 Triangle T(n,k) = number of rational (0,1) matrices of rank k (n >= 0, 0 <= k <= n). 3

%I

%S 1,1,1,1,9,6,1,49,288,174,1,225,6750,36000,22560,1,961,118800,3159750,

%T 17760600,12514320,1,3969,1807806,190071000,5295204600,34395777360,

%U 28836612000,1,16129,25316928,9271660734,1001080231200,32307576315840

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

%C Rows add to 2^(n^2).

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

%C Table 3 from M. Zivkovic, Classification of small (0,1) matrices (see link). - _Vladeta Jovovic_, Mar 28 2006

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

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

%H M. Zivkovic, <a href="http://arXiv.org/abs/math.CO/0511636">Classification of small (0,1) matrices</a>, Linear Algebra and its Applications, 414 (2006), 310-346.

%e 1; 1,1; 1,9,6; 1,49,288,174; ...

%o (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,",","; ")));) }

%Y Cf. A064231, A000409, A000410, A046747.

%K nonn,nice,tabl

%O 0,5

%A _N. J. A. Sloane_, Sep 23 2001

%E More terms and PARI code from _Michael Somos_, Sep 25, 2001

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

%E More terms from _Vladeta Jovovic_, Mar 28 2006

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Last modified November 15 19:54 EST 2018. Contains 317240 sequences. (Running on oeis4.)