Enumerating the class of minimally path connected simplicial complexes

Abstract: In 1983 Kalai proved an incredible generalisation of Cayley's formula for the number of trees on a labelled vertex set to a formula for a class of $r$-dimensional simplicial complexes. These simplicial complexes generalise trees by means of being homologically $\Bbb Q$-acyclic. In this text we consider a different generalisation of trees to the class of pure dimensional simplicial complexes that \textit{minimally connect a vertex set} (in the sense that the removal of any top dimensional face disconnects the complex). Our main result provides an upper and lower bound for the number of these minimally connected complexes on a labelled vertex set. We also prove that they are potentially vastly more topologically complex than the generalisation of Kalai. As an application of our bounds we compute the threshold probability for the connectivity of a random simplicial complex.