Bijection between trees in Stanley character formula and factorizations of a cycle

Abstract

Stanley and F\'eray gave a formula for the irreducible character of the symmetric group related to a multi-rectangular Young diagram. This formula shows that the character is a polynomial in the multi-rectangular coordinates and gives an explicit combinatorial interpretation for its coefficients in terms of counting certain decorated maps (i.e., graphs drawn on surfaces). In the current paper we concentrate on the coefficients of the top-degree monomials in the Stanley character polynomial, which corresponds to counting certain decorated plane trees. We give an explicit bijection between such trees and minimal factorizations of a cycle.