Kalai, G. Israel J. Math. (1983) 45: 337. doi:10.1007/BF02804017
Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely (kn−1)k-faces, (3)Hk(C)=0. We prove that for,Hk−1(C) is a finite group, and our main result is:. This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.