  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
$n\leq 16$ darts.
