Quadratic Coefficients of Goulden–Rattan Character Polynomials

Abstract

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).