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 (proved 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.

中文翻译：

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具有规定度数的星座图和二分图的简单公式

摘要 我们获得了简单的二次递推公式，用于计算具有指定度数（特别是 $2k$-角度）和星座的表面上的二分图。这些公式是计算这些数字的最快已知方法。我们的工作是先前关于可积层次结构（2-Toda 和 KP）的自然延伸，即 Hurwitz 数的 Pandharipande 递归（由 Okounkov 证明并由 Dubrovin-Yang-Zagier 简化），以及几个模型的公式地图（Goulden-Jackson、Carrell-Chapuy、Kazarian-Zograf）。至于那些公式，还有待寻找双射解释。我们还包括一个以相同方式导出的单调简单 Hurwitz 数的公式。这些公式在作者与 T 的后续工作中也发挥了关键作用。