This post is a sequel of [Diameter of symmetric group](https://mathoverflow.net/q/102737/34538).


Let $\Sigma$ a generating subset of $S\_n$, $\Gamma(S\_n, \Sigma)$ the Cayley graph and $d\_{\Sigma}$ the diameter of $\Gamma(S\_n, \Sigma)$.  

Let $s\_n = min\_{\Sigma}(\vert \Sigma \vert \times d\_{\Sigma})$. 


*Question:* What's the asymptotic of $s\_n$ (or the best conjecture for that)?



Asymptotic of min(#generators times diameter), for a Cayley graph of Sn
For a generating subset $S\subset S\_n$ of size $d$, the $n$-ball has size $\le (2d)^n$, which implies that the diameter $D(S)$ satisfies $D\ge\log\_{2d}(n!)\simeq n\log n$. 


So $|S|D(S)\ge \frac{d}{\log d}\log(n!)\ge \log(n!)\sim n\log n$, and thus $s\_n=\min\_S |S|D(S)\succeq n\log n$.


This is attained, according to Theorem 7 in [this paper by Kassabov](http://www.ams.org/journals/era/2005-11-06/S1079-6762-05-00146-0/), which implies that there exists $\ell$ and generating subsets $F\_n\subset S\_n$ with $D(F\_n)\preceq n\log n$ and $|F\_n|\le \ell$, implying $s\_n\preceq n\log n$. 


