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A085656 Number of positive-definite real {0,1} n X n matrices. 9
1, 3, 27, 681, 43369, 6184475, 1688686483, 665444089745 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A real matrix M is positive-definite if x M x' > 0 for all nonzero real vectors x. Equivalently, all eigenvalues of M + M' are positive.
M need not be symmetric. For the number of different values of M + M' see A085657. - Max Alekseyev, Dec 13 2005
LINKS
Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematica, Positive Definite Matrix
EXAMPLE
For n = 2 the three matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.
MATHEMATICA
Table[Count[Tuples[{0, 1}, {n, n}], _?PositiveDefiniteMatrixQ], {n, 4}] (* Eric W. Weisstein, Jan 03 2021 *)
PROG
(PARI) { a(n) = M=matrix(n, n, i, j, 2*(i==j)); r=0; b(1); r } { b(k) = local(t); if(k>n, t=0; for(i=1, n, for(j=1, i-1, if(M[i, j]==1, t++); )); r+=2^t; return; ); forvec(x=vector(k-1, i, [0, 1]), for(i=1, k-1, M[k, i]=M[i, k]=x[i]); if( matdet(vecextract(M, 2^k-1, 2^k-1), 1)>0, b(k+1) ) ) } (Alekseyev)
CROSSREFS
Cf. A055165, which counts nonsingular {0, 1} matrices and A085506, which counts {-1, 0, 1} matrices with positive eigenvalues.
Cf. A085657, A085658, A086215, A038379 (positive semi-definite matrices), A080858, A083029.
Sequence in context: A185037 A185237 A099084 * A113100 A038379 A047656
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Jul 12 2003
EXTENSIONS
More terms from Max Alekseyev, Dec 13 2005
STATUS
approved

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Last modified March 18 22:09 EDT 2024. Contains 370951 sequences. (Running on oeis4.)