A085656 Number of positive-definite real {0,1} n X n matrices.

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

Offset: 1

Author

N. J. A. Sloane, Jul 12 2003

Keywords

• nonn
• nice

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

Examples

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

Links

• Eric Weisstein's World of Mathematics, (0,1)-Matrix
• Eric Weisstein's World of Mathematica, Positive Definite Matrix
• Index entries for sequences related to binary matrices

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.

Programs

Mathematica

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

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)

Extensions

More terms from Max Alekseyev, Dec 13 2005