Séminaire Lotharingien de Combinatoire, 82B.48 (2019), 12 pp.
Simple formulas for constellations and bipartite maps with prescribed degrees
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.
Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.
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