Enumeration and randomized constructions of hypertrees

Over 30 years ago, Kalai proved a beautiful d-dimensional analog of Cayley's formula for the number of n-vertex trees. He enumerated d-dimensional hypertrees weighted by the squared size of their (d − 1)-dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of d-hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of d-hypertrees. In addition, we study a random 1-out model of d-complexes where every (d − 1)-dimensional face selects a random d-face containing it, and show that it has a negligible d-dimensional homology.