Many regular triangulations and many polytopes

Abstract We show that for fixed $$d>3$$ d > 3 and n growing to infinity there are at least $$(n!)^{d-2 \pm o(1)}$$ ( n ! ) - 2 ± o 1 different labeled combinatorial types of d -polytopes with vertices. This is about the square previous best lower bounds. As an intermediate step, we certain neighborly polytopes (such as particular realizations cyclic polytopes) have $$(n!)^{ \left\lfloor {(d-1)/2} \right\rfloor / regular triangulations.