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 https://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 https://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!