Presentation theory of symmetric groups

This article discusses the presentations used for symmetric groups, specifically, for symmetric groups on finite sets.

Coxeter presentations: using transpositions as a generating set
Suppose $$G$$ is the symmetric group on the set $$\{ 1,2,3, \dots, n+1 \}$$. Then, $$G$$ is isomorphic to a Coxeter group, with the following Coxeter presentation:

$$G \cong \langle s_i \mid s_i^2 = 1 \forall 1 \le i \le n, (s_is_{i+1})^3 = 1 \forall 1 \le i \le n - 1, (s_is_j)^2 = 1 \forall 1 \le i < j \le n, |i - j| > 1\rangle$$.

This presentation of the symmetric group on $$n+1$$ letters has $$n$$ generators and $$n(n+1)/2$$ relations.