A Family of Combinatorial Solutions to the Kp Hierarchy

We give a new explicit solution to the KP hierarchy. This is written in terms of Schur symmetric functions, and uses the known characterization of solutions to the KP hierarchy in terms of solutions to the Plücker relations. Our solution to the Pluc̈ker relations involves a countable set of variables for content, a combinatorial parameter for partitions (which themselves arise because they index the Schur functions). By specializing the content variables, we obtain a number of solutions to the KP hierarchy, including Okounkov’s result for the double Hurwitz series. Another specialization gives the m-hypermap series, which contains the double Hurwitz series as the leading coefficient. In turn this specializes to the series for hypermaps and maps in an orientable surface. For the latter series, we use one of the KP equations to obtain a remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces.