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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
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<n, ", ", "; "))); ) }
CROSSREFS
Main diagonal gives A055165.
Sequence in context: A199082 A358644 A220669 * A286331 A363036 A354741
KEYWORD
nonn,nice,tabl
AUTHOR
N. J. A. Sloane, Sep 23 2001
EXTENSIONS
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
STATUS
approved

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Last modified May 29 00:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)