I have a pretty simple question for which I was not able to find a so simple answer.

**Introduction**

I was playing around with some of the mathematical objects that can be enumerated by Catalan numbers. One of them are the famous strings of correctly matched parentheses. 

Let $P^{(n)}$ be the set containing all correctly matched parentheses strings of length $2n$, and imagine there exists a bijective funcion $f$ mapping each of the strings $s^{(n)}$ in $P^{(n)}$ to the set of integers ${1, 2, ..., n}$ such that each string of length $n$ can be denoted as $s_i^{(n)}$ where $i=f(s)$.

Define now our function of interest $G_i^n(k)$ as a function acting on the $k$-th parenthesis of $s_i^{(n)}$ the following way: 

· If the $k$-th parentheses is "(", it takes the value $0$.

· If the $k$-th parentheses is ")", it takes the number of already matched "(" characters in the string to the closest **unmatched** "(" to its left plus one. 

This can also be described by the following algorithm:

1. Look at the parentheses in the $k$-th position. If it is a "(", you output $0$ and the algorithm is finished. Otherwise, you continue to $2$.

2. Move one position to the left and check the parentheses there. If it is a ")", you do nothing and return to step $2$. If it is an already matched "(", you increase the output of the function by $1$ and return to step $2$. If it is an unmatched "(", you increase the output of the function by $1$, mark that parentheses as already matched and exit the algorithm by outputing the accumulated value of the funcion

While this may seem a bit of a mess, it is easy to see with a few examples. Focusing on two of the $n=3$ cases, the string:

$$ (()())$$

would produce the following resulting values:

$$(0,0,1,0,1,3)$$

And, for the string:

$$()(())$$

the function would take values:

$$(0,1,0,0,1,2)$$

I apologize if the notation is not precise or non-standard in any way.

My main question is just what it is known about this function and whether it has a closed form.

**My work**

After spending some time on it, I was able to identify my original problem both with Dyck paths and with full rooted binary trees. Regarding the former, $G$ can be seen as the "width" of the Dyck's path peak as seen from right to left from the $k$-th point divided by $2$. The previous examples can be expressed (using the conventional bijection of taking "(" to a step $(1,1)$ and ")" to a step $(1,-1)$) as:

[![path 1][1]][1]

And:

[![path 2][2]][2]

Then, the function I am studying can also be expressed in terms of the heights and $x$ coordinate of the peaks and valleys of the path to the left side of the $k$-th point.

**The question**

I just want to know whether this function has already been studied or if there's any closed form for it. I would also appreciate if anyone could point me to useful references regarding this topic, since most of the bibliography I could find tends to focus on counting the number of Dyck paths following certain specific patterns rather than what I am looking for here.

Thank you very much in advance!


  [1]: https://i.sstatic.net/aSUNl.png
  [2]: https://i.sstatic.net/TQmuR.png