Quadratic coefficients of Goulden-Rattan character polynomials

TL;DR

This paper proves a special case of the Goulden-Rattan positivity conjecture, demonstrating that the quadratic coefficient $C_2^2$ in character polynomials is positive, using bijections involving maps on surfaces.

Contribution

It establishes the positivity of the quadratic coefficient in Goulden-Rattan polynomials for symmetric group characters, a previously unproven special case.

Findings

Quadratic coefficient $C_2^2$ is positive.

Bijections involving maps are used in the proof.

Supports the Goulden-Rattan positivity conjecture.

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).