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A064231
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Triangle read by rows: T(n,k) = number of rational (+1,-1) matrices of rank k (n >= 1, 1 <= k <= n).
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4
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2, 8, 8, 32, 288, 192, 128, 6272, 36864, 22272, 512, 115200, 3456000, 18432000, 11550720, 2048, 1968128, 243302400, 6471168000, 36373708800, 25629327360, 8192, 32514048, 14809546752, 1557061632000, 43378316083200, 281770208133120
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OFFSET
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1,1
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COMMENTS
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Komlos and later Kahn, Komlos and Szemeredi show that almost all such matrices are invertible.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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2; 8,8; 32,288,192; 128,6272,36864,22272; ...
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PROG
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(PARI) T=matrix(4, 4); { for(n=1, 4, mm=matrix(n, n); for(k=1, n, T[n, k]=0); forvec(x=vector(n*n, i, [0, 1]), for(i=1, n, for(j=1, n, mm[i, j]=(-1)^x[i+n*(j-1)])); T[n, matrank(mm)]++); for(k=1, n, print1(T[n, k], if(k<n, ", ", "; "))); ) }
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Offset changed to 1 by T. D. Noe, Mar 02 2011
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STATUS
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approved
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