  The KP hierarchy is a completely integrable system of quadratic, partial
differential equations that generalizes the KdV hierarchy. A linear combination
of Schur functions is a solution to the KP hierarchy if and only if its
coefficients satisfy the Plucker relations from geometry. We give a solution to
the Plucker relations involving products of variables marking contents for a
partition, and thus give a new proof of a content product solution to the KP
hierarchy, previously given by Orlov and Shcherbin. In our main result, we
specialize this content product solution to prove that the generating series
for a general class of transitive ordered factorizations in the symmetric group
satisfies the KP hierarchy. These factorizations appear in geometry as
encodings of branched covers, and thus by specializing our transitive
factorization result, we are able to prove that the generating series for two
classes of branched covers satisfies the KP hierarchy. For the first of these,
the double Hurwitz series, this result has been previously given by Okounkov.
The second of these, that we call the m-hypermap series, contains the double
Hurwitz series polynomially, as the leading coefficient in m. The m-hypermap
series also specializes further, first to the series for hypermaps and then to
the series for maps, both in an orientable surface. For the latter series, we
apply one of the KP equations to obtain a new and remarkably simple recurrence
for triangulations in a surface of given genus, with a given number of faces.
This recurrence leads to explicit asymptotics for the number of triangulations
with given genus and number of faces, in recent work by Bender, Gao and
Richmond.
