  Goulden--Rattan polynomials give the exact value of the subdominant part of
the normalized characters of the symmetric groups in terms of certain
quantities ($C_i$) which describe the macroscopic shape of the Young diagram.
The Goulden--Rattan positivity conjecture states that the coefficients of these
polynomials are positive rational numbers with small denominators. We prove a
special case of this conjecture for the coefficient of the quadratic term
$C_2^2$ by applying certain bijections involving maps (i.e., graphs drawn on
surfaces).
