On July 10, 2003, Eric Weisstein computed the numbers of (0,1)-matrices all of
whose eigenvalues are real and positive, obtaining
counts for ,
2, ... of 1, 3, 25, 543, 29281, .... Based on agreement with OEIS A003024,
Weisstein then conjectured that is equal to the number of labeled acyclic
digraphs on
vertices.
This result was subsequently proved by McKay et al. (2003, 2004).
McKay, B. D.; Oggier, F. E.; Royle, G. F.; Sloane, N. J. A.; Wanless, I. M.; and Wilf, H. "Acyclic Digraphs
and Eigenvalues of -Matrices." 28 Oct 2003. http://arxiv.org/abs/math/0310423.McKay,
B. D.; Royle, G. F.; Wanless, I. M.; Oggier, F. E.; Sloane, N. J. A.;
and Wilf, H. "Acyclic Digraphs and Eigenvalues of -Matrices." J. Integer Sequences7,
Article 04.3.3, 1-5, 2004. http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Sloane/sloane15.html.Sloane,
N. J. A. Sequence A003024/M3113
in "The On-Line Encyclopedia of Integer Sequences."