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 (this is not a bijection!).
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 $|123\circ K\circ S$. This is, $$\sum_{\pi \in\mathfrak S_n} q^{|1|23(\pi)} = \sum_{\pi\in\mathfrak S_n}q^{|123 \circ K\circ S(\pi)}.$$
Why would this be the case? A bijective argument (i.e., a bijection on $\mathfrak S_n$ sending $|1|23$ to $|123\circ K\circ S$) might be especially nice!
Refinement:
Let
- $s_{54}(\pi)$ be $\pi(1)$, http://www.findstat.org/St000054,
- $s_7(\pi)$ be the number of right-to-left maxima of $\pi$, http://www.findstat.org/St000007,
let
- $s_{441}(\pi)$ be the number of indices $i$ with $\pi(i+1)=\pi(i)+1$, http://www.findstat.org/St000441
- $s_{237}(\pi)$ be the number of indices $i$ with $\pi(i)=i+1$, http://www.findstat.org/St000237,
and let
- $s_{542}(\pi)$ be the number of left-to-right minima of $\pi$, http://www.findstat.org/St000542
- $s_{991}(\pi)$ be the number of right-to-left minima of $\pi$, http://www.findstat.org/St000991
Then, apparently (checked for $n\leq8$),
$$\sum_{\pi \in\mathfrak S_n} q^{|1|23(\pi)} p^{s_{54}(\pi)}x^{s_{441}(\pi)} y^{s_{542}(\pi)}= \sum_{\pi\in\mathfrak S_n}q^{|123 \circ K\circ S(\pi)}p^{s_7\circ K\circ S(\pi)}x^{s_{237}(\pi)} y^{s_{991}(\pi)}.$$
