The permutation group $S_n$ can be generated by an element $$(i \; \; i+1)$$ together with an $n$-cycle.
First, I would like to check that the above statement is indeed true?
My main question is, is it known what the relations are when we generate $S_n$ like this?
Well, let's say that first one is on you. For the other one, one can (somewhat easily) find out that $$ \langle t, c \,|\,t^2, c^n, (tc)^{n-1}, [t, c]^3, [t, c^k]^2 = 1 \text{ for } 2 \leq k \leq n/2\rangle $$
is indeed the presentation you need. The best (to my knowledge) way to check its validity is first writing down Coxeter presentation as a reflection group, and then eliminating extra generators replacing them by products of conjugates of a transposition.