Rewriting simple permutation

Question

I am learning about the symmetric and permutations and I have gotten a bit confused about the cyclic notation. From what I read in my lecture book, I would assume that we can rewrite $(12)(13)$ as $(132)$, since $$ \begin{aligned} (12)(13)[1]=(12)[3]=3\\ (12)(13)[2]=(12)[2]=1\\ (12)(13)[3]=(12)[1]=2 \end{aligned} $$ However, Wolfram Alpha has a function called 'permutation cycles' which supposedly can simplify cycles, and it tells me that $(12)(13)=(123)$. Have I misunderstood something?

Answer 1

Permutations are evaluated from left to right meaning first $(12)$ and then $(13)$. It is exponential notation, not functional:

$$1^{(12)(13)} = 2^{(13)} = 2$$ $$2^{(12)(13)} = 1^{(13)} = 3$$ $$3^{(12)(13)} = 3^{(13)} = 1.$$