Simple Formulas for Constellations and Bipartite Maps with Prescribed Degrees

Abstract

AbstractWe obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, $2k$-angulations) and constellations. These formulas are the fastest known way of computing these numbers.Our work is a natural extension of previous works on integrable hierarchies (2-Toda and KP), namely, the Pandharipande recursion for Hurwitz numbers (proved by Okounkov and simplified by Dubrovin–Yang–Zagier), as well as formulas for several models of maps (Goulden–Jackson, Carrell–Chapuy, Kazarian–Zograf). As for those formulas, a bijective interpretation is still to be found. We also include a formula for monotone simple Hurwitz numbers derived in the same fashion.These formulas also play a key role in subsequent work of the author with T. Budzinski establishing the hyperbolic local limit of random bipartite maps of large genus.