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Number of positive-definite real {0,1} n X n matrices.
9

%I #14 Jan 11 2021 22:36:09

%S 1,3,27,681,43369,6184475,1688686483,665444089745

%N Number of positive-definite real {0,1} n X n matrices.

%C 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.

%C M need not be symmetric. For the number of different values of M + M' see A085657. - _Max Alekseyev_, Dec 13 2005

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/01-Matrix.html">(0,1)-Matrix</a>

%H Eric Weisstein's World of Mathematica, <a href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html">Positive Definite Matrix</a>

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%e For n = 2 the three matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.

%t Table[Count[Tuples[{0, 1}, {n, n}], _?PositiveDefiniteMatrixQ], {n, 4}] (* _Eric W. Weisstein_, Jan 03 2021 *)

%o (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)

%Y Cf. A055165, which counts nonsingular {0, 1} matrices and A085506, which counts {-1, 0, 1} matrices with positive eigenvalues.

%Y Cf. A085657, A085658, A086215, A038379 (positive semi-definite matrices), A080858, A083029.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_, Jul 12 2003

%E More terms from _Max Alekseyev_, Dec 13 2005