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

TL;DR

This paper establishes a bijection between decorated plane trees and minimal factorizations of a cycle, providing a combinatorial interpretation for specific coefficients in Stanley's character formula.

Contribution

It introduces a new explicit bijection linking decorated trees to cycle factorizations, enhancing understanding of Stanley's character polynomial coefficients.

Findings

Bijection between decorated plane trees and cycle factorizations

Explicit combinatorial interpretation of top-degree coefficients

Connection between graph embeddings and algebraic factorizations

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.