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

%I A064230 #21 Jul 03 2025 17:43:51
%S A064230 1,1,1,1,9,6,1,49,288,174,1,225,6750,36000,22560,1,961,118800,3159750,
%T A064230 17760600,12514320,1,3969,1807806,190071000,5295204600,34395777360,
%U A064230 28836612000,1,16129,25316928,9271660734,1001080231200,32307576315840
%N A064230 Triangle T(n,k) = number of rational (0,1) matrices of rank k (n >= 0, 0 <= k <= n).
%C A064230 Rows add to 2^(n^2).
%C A064230 Komlos and later Kahn, Komlos and Szemeredi show that almost all such matrices are invertible.
%C A064230 Table 3 from M. Zivkovic, Classification of small (0,1) matrices (see link). - _Vladeta Jovovic_, Mar 28 2006
%D A064230 J. Kahn, J. Komlos and E. Szemeredi: On the probability that a random +-1 matrix is singular, J. AMS 8 (1995), 223-240.
%D A064230 J. Komlos, On the determinants of random matrices, Studia Sci. Math. Hungar., 3 (1968), 387-399.
%H A064230 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.
%F A064230 Sum_{k=1..n} k * T(n,k) = A086875(n). - _Alois P. Heinz_, Jun 18 2022
%e A064230 Triangle T(n,k) begins:
%e A064230   1;
%e A064230   1,   1;
%e A064230   1,   9,      6;
%e A064230   1,  49,    288,     174;
%e A064230   1, 225,   6750,   36000,    22560;
%e A064230   1, 961, 118800, 3159750, 17760600, 12514320;
%e A064230   ...
%o A064230 (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 A064230 Cf. A064231, A000409, A000410, A046747, A086875.
%Y A064230 Main diagonal gives A055165.
%K A064230 nonn,nice,tabl
%O A064230 0,5
%A A064230 _N. J. A. Sloane_, Sep 23 2001
%E A064230 More terms and PARI code from _Michael Somos_, Sep 25 2001
%E A064230 6 more terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Dec 17 2004
%E A064230 More terms from _Vladeta Jovovic_, Mar 28 2006