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 A064230 Triangle T(n,k) = number of rational (0,1) matrices of rank k (n >= 0, 0 <= k <= n). 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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