Hurwitz enumeration problem through the perspective of Cayley graph

The Hurwitz enumeration problem studies how to determine the Hurwitz number for a branch profile, which counts the number of ways a per mutation can be factored into transpositions. In this paper, we consider the length of strictly monotone factorization from the perspective of Cayley graph theory. We represent the factorization problem using a Cayley graph, where vertices are permutations and edges are transpositions. Our focus is proving the unique monotone factorization theorem, which states that for a given permutation, there is only one monotone factorization of minimal length. To prove this, we employ inductive arguments on the structure of the Cayley graph. The key insight is using the connectivity of the graph and properties of shortest paths to characterize the uniqueness of the minimal factorization. This inductive approach allows us to rigorously connect the combinatorial Hurwitz problem to foundational graph concepts. Overall, this paper makes important theoretical advances in enumerating Hurwitz numbers by using Cayley graphs and induction to prove the novel unique monotone factorization theorem. The connections drawn between combinatorics, graphs, and inductive proofs are technically innovative. This theoretical foundation will hopefully stimulate further research into the deep links between the Hurwitz problem and other branches of mathematics.