Hodge integrals, Hurwitz numbers, and Symmetric Groups

TL;DR

This paper proves combinatorial results related to a conjectured formula on Hodge integrals, which are crucial for its proof and applications, and also derives closed expressions for generating series of Hurwitz numbers and related integrals.

Contribution

It provides new combinatorial proofs and closed-form expressions for Hodge integrals and Hurwitz numbers, advancing understanding in algebraic geometry and enumerative combinatorics.

Findings

Proved combinatorial results supporting the Mariño-Vafa formula

Derived closed expressions for generating series of Hurwitz numbers

Enhanced understanding of the relationship between Hodge integrals and Hurwitz numbers

Abstract

We prove some combinatorial results related to a formula on Hodge integrals conjectured by Mari\~no and Vafa. These results play important roles in the proof and applications of this formula by the author jointly with Chiu-Chu Melissa Liu and Kefeng Liu. We also compare with some related results on Hurwitz numbers and obtain some closed expressions for the generating series of Hurwitz numbers and the related Hodge integrals.