Polynomiality of monotone Hurwitz numbers in higher genera

TL;DR

This paper extends the understanding of monotone Hurwitz numbers to higher genera, providing explicit formulas and demonstrating their polynomial nature, which parallels classical Hurwitz number properties.

Contribution

It introduces explicit formulas for monotone Hurwitz numbers in genus one and arbitrary positive genus, and proves their polynomiality in higher genera.

Findings

Explicit formula for genus one monotone Hurwitz numbers

Generating function form for arbitrary genus

Polynomiality of monotone Hurwitz numbers in higher genera

Abstract

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e., up to a specified combinatorial factor, the monotone Hurwitz number in genus g with ramification specified by a given partition is a polynomial indexed by g in the parts of the partition.