Standard cycle form and Foata's fundamental transformation

Let $\sigma \in S_n$. Write its disjoint-cycle decomposition (Every permutation of a finite set is a product of pairwise disjoint cycles, uniquely up to reordering and cyclic rotation) and insert each omitted fixed point as a one-cycle, as permitted by the cycle-type convention of Support, fixed points, disjoint cycles, cycle length, disjoint-cycle decompositions, and cycle type. Then rewrite each cycle so that its largest entry appears first and order the cycles by increasing first entry. The result is the standard cycle form of $\sigma$.

Deleting the parentheses from that ordered list of cycles produces a word of length $n$ whose entries are $0,\dots ,n-1$ each once. The corresponding permutation in one-line notation is Foata's fundamental transformation $\mathcal{F}(\sigma )$.

Example: if $\sigma =(031)(2)(45)$, then the standard cycle form is $(2)(310)(54)$ and $\mathcal{F}(\sigma )=[2,3,1,0,5,4]$.