# Simple formulas for constellations and bipartite maps with prescribed degrees

## TL;DR

This paper introduces simple quadratic recurrence formulas for counting bipartite maps and constellations with prescribed degrees, providing the fastest known computational method and extending previous integrable hierarchy results.

## Contribution

It presents new quadratic recurrence formulas for bipartite maps and constellations, extending integrable hierarchy approaches and improving computational efficiency.

## Key findings

- Formulas for bipartite maps with prescribed degrees
- Fastest known computation method for these counts
- Application to hyperbolic local limits of large genus maps

## Abstract

We 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 (proven 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.