Cycle factorizations and one-faced graph embeddings

TL;DR

This paper introduces a polynomial encoding of transposition factorizations of n-cycles, leading to a formula for counting one-faced graph embeddings, bridging algebraic factorizations with topological graph embeddings.

Contribution

It provides a novel polynomial expression for transposition factorizations and derives a closed-form formula for counting one-faced graph embeddings.

Findings

Derived a closed-form expression for the polynomial sum over factorizations

Established a formula for counting 1-faced graph embeddings

Connected algebraic factorizations with topological graph properties

Abstract

Consider factorizations into transpositions of an n-cycle in the symmetric group S_n. To every such factorization we assign a monomial in variables w_{ij} that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we deduce a formula for the number of 1-faced embeddings of a given graph.