Quadratic Coefficients of Goulden–Rattan Character Polynomials

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).