In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on n darts, thus solving an analogue of Tutte’s problem in dimension three. The generating series we derive also counts free subgroups of index n in \(\Delta ^+ = {\mathbb {Z}}_2*{\mathbb {Z}}_2*{\mathbb {Z}}_2\) via a simple bijection between pavings and finite index subgroups which can be deduced from the action of \(\Delta ^+\) on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphism classes of pavings, which correspond to conjugacy classes of free subgroups of finite index in \(\Delta ^+\). Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on \(n\le 16\) darts.

在本文中，我们在n 个飞镖上导出了称为铺路或三维地图的细胞复合体数量的生成级数，从而解决了三维图特问题的类似问题。我们推导的生成级数还通过铺路和有限索引子群之间的简单双射，可以从\(\Delta ^+\)对给定子群的陪集的作用推导出来。然后我们证明这个生成级数是非完整的。此外，我们提供并研究了铺路同构类的生成级数，其对应于\(\Delta ^+\)中有限索引的自由子群的共轭类。使用作者设计的软件进行的计算实验提供了一些有关\(n\le 16\)飞镖铺路的拓扑和组合的统计数据。