Eric W. Weisstein of Wolfram Research, Inc., conjectured that the number of
real n x n matrices of 0's and 1's all of whose eigenvalues are real and
positive coincides with number of acyclic digraphs with n labeled vertices.
This is proved in
B. D. McKay, F. E. Oggier, G. F. Royle,
N. J. A. Sloane, Ian M. Wanless and Herbert S. Wilf.
"Acyclic digraphs and eigenvalues of (0,1)-matrices",
Journal of Integer Sequences, vol.7, August 2004
(arXiv: math.CO/0310423)