A085506 Number of {-1,0,1} n X n matrices that have (real and) positive eigenvalues.

1, 5, 133, 18905

Offset: 1

Author

Eric W. Weisstein, Jul 02 2003

Keywords

• nonn
• more

Links

• B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, J. Integer Sequences, 7 (2004), #04.3.3.
• B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, arXiv:math/0310423 [math.CO], 2003.
• Eric Weisstein's World of Mathematics, Positive Definite Matrix
• Index entries for sequences related to binary matrices

Programs

Mathematica

a[n_] := Module[{M, iter, cnt = 0}, M = Table[a[i, j], {i, 1, n}, {j, 1, n}]; iter = Thread[{Flatten[M], -1, 1}]; Do[If[AllTrue[Eigenvalues[M], If[Im[#] != 0, False, Positive[#]]&], cnt++], Evaluate[Sequence @@ iter]]; cnt];
Do[Print[n, " ", a[n]], {n, 1, 3}] (* Jean-François Alcover, Dec 09 2018 *)

Extensions

Offset corrected by Max Alekseyev, Oct 18 2008