Quadratic Coefficients of Goulden–Rattan Character Polynomials

Abstract

AbstractGoulden–Rattan polynomials give the exact value of the subdominant part of the normalized characters of the symmetric groups in terms of certain quantities $$(C_i)$$ ( 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 $$C_2^2$$ C 2 2 by applying certain bijections involving maps (i.e., graphs drawn on surfaces).