Simple formulas for constellations and bipartite maps with prescribed degrees
arXiv:1904.05371 · doi:10.4153/S0008414X19000555
Abstract
We obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, -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.
16 pages, 4 figures, extended abstract in FPSAC 2019, to appear in CJM