Let $\sigma \in S_4$ be the permutation that sends $(a,b,c,d) \to (b,d,a,c).$ Then the cycle notation is $(1342)=(23)(34)(12).$
But if we denote $a$ as $1,$ $b$ as $2$, $c$ as $3$ and $d$ as $4,$ then it means $\sigma(1)=2, \sigma(2)=4, \sigma(3)= 1$ and $\sigma(4)=3.$ Hence, isnt the cycle notation $(1243)$ instead?
Kindly advise, thank you.