Combinatorics of the Berezin-Karpelevich Integral

The Berezin-Karpelevich integral is a double integral over unitary matrices which plays the role of the Itzykson-Zuber integral in rectangular matrix models. We obtain a topological expansion of the Berezin-Karpelevich integral in terms of monotone Hurwitz numbers, and obtain from this certain combinatorial identities.