Monotone Hurwitz numbers in genus zero

TL;DR

This paper introduces and analyzes monotone Hurwitz numbers, a restricted class of branched covers, providing explicit formulas in genus zero and a differential equation characterizing their generating function.

Contribution

It establishes the monotone join-cut equation and derives an explicit formula for genus zero monotone Hurwitz numbers, advancing understanding of their combinatorial structure.

Findings

Derived the monotone join-cut differential equation.

Provided an explicit formula for genus zero monotone Hurwitz numbers.

Connected monotone Hurwitz numbers to asymptotic analysis of matrix integrals.

Abstract

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of the branched covers counted by the Hurwitz numbers, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.