  Recently, B. Yang and J. Yang derived a family of rational solutions to the
Sasa-Satsuma equation, and showed that any of its members constitutes a
partial-rogue wave provided that an associated generalised Okamoto polynomial
has no real roots or no imaginary roots. In this paper, we derive exact
formulas for the number of real and the number of imaginary roots of the
generalised Okamoto polynomials. On the one hand, this yields a list of
partial-rogue waves that satisfy the Sasa-Satsuma equation. On the other hand,
it gives families of rational solutions of the fourth Painlev\'e equation that
are pole-free on either the real line or the imaginary line. To obtain these
formulas, we develop an algorithmic procedure to derive the qualitative
distribution of singularities on the real line for real solutions of Painlev\'e
equations, starting from the known distribution for a seed solution, through
the action of B\"acklund transformations on the rational surfaces forming their
spaces of initial conditions.
