This is (at least for now) a question out of curiosity, there is no "deeper" meaning to it I know of. In fact, my main question is: is the observation below obvious?

To state the observation I have to define two statistics on permutations $|1|23:\mathfrak S_n\to \mathbb N$ and $|123:\mathfrak S_n\to \mathbb N$, and two maps, $K:\mathfrak S_n\to\mathfrak S_n$ and $S:\mathfrak S_n\to\mathfrak S_n$.

Let $\pi$ be a permutation, then an occurrence of the vincular pattern $|1|23$ (warning: notations vary) is an occurrence of the ordinary pattern $123$ such that the first matched entries are the first two entries of the permutation. In other words the number of occurrences of $|1|23$ in $\pi$ is zero, if the $\pi(2) < \pi(1)$, and it is the number of entries larger than $\pi(2)$ otherwise. The statistic http://findstat.org/St001084 counts the number of occurrences of $|1|23$ in $\pi$.

Similarly, an occurrence of the vincular pattern $|123$ is an occurrence of the ordinary pattern $123$ such that the first matched entry is the first entry of the permutation. The statistic http://findstat.org/St000804 counts the number of occurrences of $|123$ in $\pi$.

Now, for the maps! Let $K$ be the inverse Kreweras complement http://findstat.org/Mp00089 mapping $\pi$ to $(1,\dots,n)\pi^{-1}$, and let $S$ be the Simion-Schmitt http://findstat.org/Mp00068 map, sending any permutation to a $123$ avoiding permutation.

Observation:

At least for $n\leq 8$, the distribution over $\mathfrak S_n$ of the number of occurrences of $|1|23$ is the same as the distribution of $|123\circ K\circ S$.

Why would this be the case? A bijective argument might be especially nice!