## Permutation and cycles

Let $$[n]$$ be the set $$\{1,2,...,n\}$$ (we call each element a vertex) and a permutation of $$[n]$$ be $$\pi=[\pi(1),\pi(2),\cdots,\pi(n)]$$, i.e., denote $$\pi(x)=y$$ the fact that in a permutation, position $$x$$ has vertex $$y$$. There are $$n!$$ possible permutations of $$[n]$$. [more]