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Simple formulas for constellations and bipartite maps with prescribed degrees
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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.
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Cited by 1 Pith paper
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Combinatorics of Bousquet-M\'elou-Schaeffer numbers in the light of topological recursion
For stable genera, Bousquet-Mélou-Schaeffer numbers factor as a product of explicit Pochhammer-type factors times a polynomial in the partition parts, and the paper proves this combinatorially.
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