What are the least sets of generators for $S_n$

Question

Nearly all the books I read give $S_n$ $(n \geq 2)$ and the generating set $\{(i,i+1) | 1 \leq i < n \}$ as an example when talking about presentation groups. But is $\{(i,i+1) | 1 \leq i < n \}$ the least set of generators, i.e., is the order of any generating set for $S_n$ equal to or greater than $n-1$? If it is the least, how to prove? Are there any other least set of generators? In general, what do these least sets look like?

Forgive me for so many questions. Thanks sincerely for any answers or hints.

Answer 1

I think $(1,2), (1,2,\ldots, n)$ is also a set of generators.

Answer 2

It is proved in

J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205.

that the probability that a random pair elements of $S_n$ generate $S_n$ approaches $3/4$ has $n \to \infty$, and the probability that they generate $A_n$ approaches $1/4$.