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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

let

and let

Then, apparently (checked for $n\leq8$),

$$\sum_{\pi \in\mathfrak S_n} q^{|1|23(\pi)} p^{s_{54}(\pi)}x^{s_{441}(\pi)} y^{s_{542}(\pi)}= \sum_{\pi\in\mathfrak S_n}q^{|123 \circ K\circ S(\pi)}p^{s_7\circ K\circ S(\pi)}x^{s_{237}(\pi)} y^{s_{991}(\pi)}.$$

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  • $\begingroup$ Just two notes: 1) Like you observe, a permutation has an occurrence of |1|23 if and only if it starts pi(1)<pi(2) and the number of occurrences are equal to the number of elements in pi that are larger than pi(2). 2) If you look at the other pattern then the number of occurrences are the number of non-inversions “appearing above” pi(1). I do not have much experience with the maps you use, so this is were I stop being of assistance, but I would start by looking at what the maps do with non-inversions, especially non-inversions “appearing above" pi(1). $\endgroup$ – Henning Arnór Úlfarsson Jan 12 '18 at 9:02

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