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A Dyck word is a sequence of open and closed brackets such that the brackets come in correctly matched pairs. For example $(()(()))()$ is a Dyck word, while $())(()$ is not. A Dyck path is a visual representation of a Dyck word, where for every open bracket the path goes up one step and for every closed bracket the path goes down one step. For example the Dyck path to $(()(()))()$ is It is easily seen that the Dyck path uniquely determines the Dyck word and that any such path, which stays above or at the starting height and ends at the starting height, is a Dyck path. A peak in a Dyck word is an occurrence of $()$.

Let $D(n,k)$ denote the set of Dyck paths/words of length $2n$ with $k$ many peaks, then the cardinality $\#D(n,k)$ is given by the Narayana numbers $N(n,k) = \frac{1}{n} {n \choose k} {n \choose k-1}$.

Question: In the asymptotic regime where $N(n,k)$ is largest (that is $\frac{k}{n} \rightarrow \frac{1}{2}$) I am interested in the limiting distribution of the peaks on the horizontal axis. For any Dyck word $w \in D(n,k)$ let $q_1(w),...,q_k(w) \in [2n-1]$ denote the positions of the peaks of $w$, i.e. such that the positions $q_i,q_{i}+1$ in $w$ are filled with $()$ for all $i \leq k$. If we rescale the interval $[0,2n]$ to $[0,1]$, we can describe a probability distribution \begin{align*} & \mu_{n,k} := \frac{1}{N(n,k)} \frac{1}{k} \sum\limits_{w \in D(n,k)} \sum\limits_{i \leq k} \delta_{\frac{1}{2n}q_i(w)} \ . \end{align*} I would be very interested in the weak limit of $\mu_{n,k}$ in the regime $\frac{k}{n} \rightarrow \frac{1}{2}$. (This assumes $1/n$ is the correct rescaling to get a non-trivial distribution as limit, of which I am not entirely sure.)

Any help is much appreciated!

A Dyck word is a sequence of open and closed brackets such that the brackets come in correctly matched pairs. For example $(()(()))()$ is a Dyck word, while $())(()$ is not. A Dyck path is a visual representation of a Dyck word, where for every open bracket the path goes up one step and for every closed bracket the path goes down one step. For example the Dyck path to $(()(()))()$ is It is easily seen that the Dyck path uniquely determines the Dyck word and that any such path, which stays above or at the starting height and ends at the starting height, is a Dyck path. A peak in a Dyck word is an occurrence of $()$.

Let $D(n,k)$ denote the set of Dyck paths/words of length $2n$ with $k$ many peaks, then the cardinality $\#D(n,k)$ is given by the Narayana numbers $N(n,k) = \frac{1}{n} {n \choose k} {n \choose k-1}$.

Question: In the asymptotic regime where $N(n,k)$ is largest (that is $\frac{k}{n} \rightarrow \frac{1}{2}$) I am interested in the limiting distribution of the peaks on the horizontal axis. For any Dyck word $w \in D(n,k)$ let $q_1(w),...,q_k(w) \in [2n-1]$ denote the positions of the peaks of $w$, i.e. such that the positions $q_i,q_{i}+1$ in $w$ are filled with $()$ for all $i \leq k$. If we rescale the interval $[0,2n]$ to $[0,1]$, we can describe a probability distribution \begin{align*} & \mu_{n,k} := \frac{1}{N(n,k)} \frac{1}{k} \sum\limits_{w \in D(n,k)} \sum\limits_{i \leq k} \delta_{\frac{1}{2n}q_i(w)} \ . \end{align*} I would be very interested in the weak limit of $\mu_{n,k}$ in the regime $\frac{k}{n} \rightarrow \frac{1}{2}$.

Any help is much appreciated!

A Dyck word is a sequence of open and closed brackets such that the brackets come in correctly matched pairs. For example $(()(()))()$ is a Dyck word, while $())(()$ is not. A Dyck path is a visual representation of a Dyck word, where for every open bracket the path goes up one step and for every closed bracket the path goes down one step. For example the Dyck path to $(()(()))()$ is It is easily seen that the Dyck path uniquely determines the Dyck word and that any such path, which stays above or at the starting height and ends at the starting height, is a Dyck path. A peak in a Dyck word is an occurrence of $()$.

Let $D(n,k)$ denote the set of Dyck paths/words of length $2n$ with $k$ many peaks, then the cardinality $\#D(n,k)$ is given by the Narayana numbers $N(n,k) = \frac{1}{n} {n \choose k} {n \choose k-1}$.

Question: In the asymptotic regime where $N(n,k)$ is largest (that is $\frac{k}{n} \rightarrow \frac{1}{2}$) I am interested in the limiting distribution of the peaks on the horizontal axis. For any Dyck word $w \in D(n,k)$ let $q_1(w),...,q_k(w) \in [2n-1]$ denote the positions of the peaks of $w$, i.e. such that the positions $q_i,q_{i+1}$ in $w$ are filled with $()$. If we rescale the interval $[0,2n]$ to $[0,1]$, we can describe a probability distribution \begin{align*} & \mu_{n,k} := \frac{1}{N(n,k)} \frac{1}{k} \sum\limits_{w \in D(n,k)} \sum\limits_{i \leq k} \delta_{\frac{1}{2n}q_i(w)} \ . \end{align*} I would be very interested in the weak limit of $\mu_{n,k}$ in the regime $\frac{k}{n} \rightarrow \frac{1}{2}$.

Any help is much apprechiated!

A Dyck word is a sequence of open and closed brackets such that the brackets come in correctly matched pairs. For example $(()(()))()$ is a Dyck word, while $())(()$ is not. A Dyck path is a visual representation of a Dyck word, where for every open bracket the path goes up one step and for every closed bracket the path goes down one step. For example the Dyck path to $(()(()))()$ is It is easily seen that the Dyck path uniquely determines the Dyck word and that any such path, which stays above or at the starting height and ends at the starting height, is a Dyck path. A peak in a Dyck word is an occurrence of $()$.

Let $D(n,k)$ denote the set of Dyck paths/words of length $2n$ with $k$ many peaks, then the cardinality $\#D(n,k)$ is given by the Narayana numbers $N(n,k) = \frac{1}{n} {n \choose k} {n \choose k-1}$.

Question: In the asymptotic regime where $N(n,k)$ is largest (that is $\frac{k}{n} \rightarrow \frac{1}{2}$) I am interested in the limiting distribution of the peaks on the horizontal axis. For any Dyck word $w \in D(n,k)$ let $q_1(w),...,q_k(w) \in [2n-1]$ denote the positions of the peaks of $w$, i.e. such that the positions $q_i,q_{i}+1$ in $w$ are filled with $()$ for all $i \leq k$. If we rescale the interval $[0,2n]$ to $[0,1]$, we can describe a probability distribution \begin{align*} & \mu_{n,k} := \frac{1}{N(n,k)} \frac{1}{k} \sum\limits_{w \in D(n,k)} \sum\limits_{i \leq k} \delta_{\frac{1}{2n}q_i(w)} \ . \end{align*} I would be very interested in the weak limit of $\mu_{n,k}$ in the regime $\frac{k}{n} \rightarrow \frac{1}{2}$. (This assumes $1/n$ is the correct rescaling to get a non-trivial distribution as limit, of which I am not entirely sure.)

Any help is much appreciated!

A Dyck word is a sequence of open and closed brackets such that the brackets come in correctly matched pairs. For example $(()(()))()$ is a Dyck word, while $())(()$ is not. A Dyck path is a visual representation of a Dyck word, where for every open bracket the path goes up one step and for every closed bracket the path goes down one step. For example the Dyck path to $(()(()))()$ is It is easily seen that the Dyck path uniquely determines the Dyck word and that any such path, which stays above or at the starting height and ends at the starting height, is a Dyck path. A peak in a Dyck word is an occurrence of $()$.

The Narayana numbers $N(n,k) = \frac{1}{n} {n \choose k} {n \choose k-1}$ count the number of Dyck paths/words of length $2n$ with $k$ many peaks.

Question: Given $q,k,n \in \mathbb{N}$ with $k \leq n$ and $q < 2n$ and a Dyck word $W$ pulled randomly (uniformly) from the set of Dyck words of length $2n$ with $k$ many peaks, what is the probability that the positions $q$ and $q+1$ form a peak in $W$?

Trivial examples: For $k=n$, the only possible Dyck word is $()()\cdots()$ and the probability is $1$, if $q$ is odd, and zero otherwise. For $k=1$, the only possible Dyck word is $((\cdots()\cdots))$ and the probability is $1$ for $q=n$ and zero otherwise.

A Dyck word is a sequence of open and closed brackets such that the brackets come in correctly matched pairs. For example $(()(()))()$ is a Dyck word, while $())(()$ is not. A Dyck path is a visual representation of a Dyck word, where for every open bracket the path goes up one step and for every closed bracket the path goes down one step. For example the Dyck path to $(()(()))()$ is It is easily seen that the Dyck path uniquely determines the Dyck word and that any such path, which stays above or at the starting height and ends at the starting height, is a Dyck path. A peak in a Dyck word is an occurrence of $()$.

Let $D(n,k)$ denote the set of Dyck paths/words of length $2n$ with $k$ many peaks, then the cardinality $\#D(n,k)$ is given by the Narayana numbers $N(n,k) = \frac{1}{n} {n \choose k} {n \choose k-1}$.

Question: In the asymptotic regime where $N(n,k)$ is largest (that is $\frac{k}{n} \rightarrow \frac{1}{2}$) I am interested in the limiting distribution of the peaks on the horizontal axis. For any Dyck word $w \in D(n,k)$ let $q_1(w),...,q_k(w) \in [2n-1]$ denote the positions of the peaks of $w$, i.e. such that the positions $q_i,q_{i+1}$ in $w$ are filled with $()$. If we rescale the interval $[0,2n]$ to $[0,1]$, we can describe a probability distribution \begin{align*} & \mu_{n,k} := \frac{1}{N(n,k)} \frac{1}{k} \sum\limits_{w \in D(n,k)} \sum\limits_{i \leq k} \delta_{\frac{1}{2n}q_i(w)} \ . \end{align*} I would be very interested in the weak limit of $\mu_{n,k}$ in the regime $\frac{k}{n} \rightarrow \frac{1}{2}$.

Any help is much apprechiated!