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Séminaire Lotharingien de Combinatoire, 82B.48 (2019), 12 pp.

# Baptiste Louf

# Simple formulas for constellations and bipartite maps with prescribed degrees

**Abstract.**
We obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, 2*k*-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.

Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.

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