Let $s_d(n,N)$ be the number of different $d$-dimensional simplicial spheres on $n$ labelled vertices and $N$ facets (= $d$-simplices). I am in search for the best know upper bounds, especially for $d\ge 3$. In particular, is there any hope for the following:
Question: Is $s_d(n,N) = O(n^{N/d})$?
But I am generally interested in any non-trivial upper bound. What I know so far:
- Counting labelled simplicial complexes with $n$ vertices and $N$ facets gives an upper bound of $\sim n^{(d+1)N}$.
- It is an open question whether there is an exponential upper bound on the number of $d$-spheres with $N$ facets.
- There has been much research into upper bounds for simplicial spheres in terms of vertex count only. For example, Kalai poved that there are at most $\smash{2^{O(n^{\lceil d/2\rceil}\log(n))}}$ $n$-vertex $d$-spheres. I am asking about a more fine grained counting, distinguishing by number of facets. Considering the lower and upper bound theorems, my hope seems justified in the extreme cases $N\sim dn$ and $N\sim n^{\lceil d/2\rceil}$.
