Foata's transformation is a bijection of $S_n$

For a word $v$ and a letter $x$, define ${\delta }_x(v)$ by the inverse block rule: if the first letter of $v$ is $\le x$, cut $v$ immediately before each later letter $\le x$; if the first letter is $>x$, cut immediately before each later letter $>x$; in each block move the first letter to the end. This undoes ${\gamma }_x$, because ${\gamma }_x$ and ${\delta }_x$ use the same distinguished letters to recover the same blocks and then reverse the same cyclic move within each block.

Define a recursive map $\mathrm{\Psi }$ by $\mathrm{\Psi }(ϵ)=ϵ$ and $\mathrm{\Psi }(vx)=\mathrm{\Psi }({\delta }_x(v))x$, where $x$ is the last letter. Induction on word length, using step 1.1 at the last recursive step, gives $\mathrm{\Psi }(\mathrm{\Phi }(w))=w=\mathrm{\Phi }(\mathrm{\Psi }(w))$ for every permutation word $w$. Hence $\mathrm{\Psi }$ is a two-sided inverse to $\mathrm{\Phi }$.

A map with a two-sided inverse is bijective by [L1], so $\mathrm{\Phi }$ is a bijection on $S_n$ for every $n$.