The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.

The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].

The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where

• $des$ is the number of descents, St000021The number of descents of a permutation.,
• $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
• $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
• $Ebot$ is the sum of the excedance bottoms,
• $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
• $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
• $Res$ is the sum of the (right) embracing numbers,
• $Ine$ is the sum of the side numbers.

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.