Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed ramification data. Certain kinds of ramification data are of particular interest, such as double Hurwitz numbers, which count covers with fixed arbitrary ramification over 0 and ∞ and simple ramification over b points, where b is given by the Riemann-Hurwitz formula. In this work, we introduce the notion of bi-pruned double Hurwitz numbers. This is a new enumerative problem, which yields smaller numbers but completely determines double Hurwitz numbers. They count a relevant subset of covers and share many properties with double Hurwitz numbers, such as piecewise polynomial behaviour and an expression in the symmetric group. Thus, we may view them as a core of the double Hurwitz numbers problem. This work is built on and generalises previous results of Zvonkine [18], Irving [13], Irving-Rattan [12], Do–Norbury [3] and the author [9].

Hurwitz数字用固定的分枝数据枚举Riemann球的分枝覆盖层。某些种类的分枝数据特别令人关注，例如双Hurwitz数，其计数覆盖了0和∞上的固定任意分枝和b点上的简单分枝，其中b由Riemann-Hurwitz公式给出。在这项工作中，我们介绍了双修剪双Hurwitz数的概念。这是一个新的枚举问题，它产生的数字较小，但完全确定了双重Hurwitz数。他们计算封面的一个相关子集，并使用双Hurwitz数共享许多属性，例如分段多项式行为和对称组中的一个表达式。因此，我们可以将它们视为双重Hurwitz数问题的核心。这项工作建立在Zvonkine [18]，Irving [13]，Irving-Rattan [12]，Do–Norbury [3]和作者[9]的先前结果之上，并对其进行概括。