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.Comment: 22 pages, submitted to the Canadian Journal of Mathematic
