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.Comment: 23 page