Explicit formulae for one-part double Hurwitz numbers with completed 3-cycles

TL;DR

This paper derives explicit formulas for a specific class of double Hurwitz numbers involving completed 3-cycles and introduces combinatorial Hodge integrals, linking algebraic geometry and combinatorics.

Contribution

It provides the first explicit formulas for one-part double Hurwitz numbers with completed 3-cycles and defines combinatorial Hodge integrals inspired by the ELSV formula.

Findings

Explicit formulas for the specified Hurwitz numbers.

Introduction of combinatorial Hodge integrals.

Derived properties and formulae for these integrals.

Abstract

We prove two explicit formulae for one-part double Hurwitz numbers with completed 3-cycles. We define "combinatorial Hodge integrals" from these numbers in the spirit of the celebrated ELSV formula. The obtained results imply some explicit formulae and properties of the combinatorial Hodge integrals.