  We derive a formula for the signature of the symmetrized Stokes matrix
$\cal{S}+\cal{S}^\mathrm{T}$ for the $tt^*$-Toda equation. As a corollary, we
verify a conjecture of Cecotti and Vafa regarding when
$\cal{S}+\cal{S}^\mathrm{T}$ is positive definite, reminiscent of a formula of
Beukers and Heckmann for the generalized hypergeometric equation. The condition
that $\cal{S}+\cal{S}^\mathrm{T}$ is positive definite is prominent in the work
of Cecotti and Vafa on the $tt^*$ equation; we show that the Stokes matrices
$\cal{S}$ satisfying this condition are parameterized by the points of an open
convex polytope.
