In recent work with Michael Albert and Nik Ruškuc, a family of words has arisen which is counted by the Catalan numbers. I've looked at Richard Stanley's Catalan exercises in EC2 a …

This is something I found in trying to work on Vince Vatter's excellent question. I have no solution, but a much more precise conjecture. Recall that a rooted planar tree is a roo …

What is the number of ways to parenthesize $n$ elements using applications of operators of arbitrary arities larger than or equal to $2$? For example, for $n=3$, there are $3$ ways …

A partition of $n$ is a weakly decreasing sequence of natural numbers $\lambda = (\lambda_1, \lambda_2, \dots)$ such that $\sum \lambda_i = n$. Its length $l(\lambda)$ is the numbe …

I had completed a paper describing the $q$-Catalan numbers, which is the $q$-analog of the Catalan numbers. The $n$-th Catalan numbers can be represented by: $$C_n=\frac{1}{n+1}{ …

Here are some beginner questions on partition algebras... I am trying to understand the monoid called $P_k$ in Tom Halverson, Arun Ram, Partition Algebras. For the sake of simplic …

From $0 = 0.5 - 0.5 = 0.5 - \sqrt{0.25}$, we can adjust the subtrahend slightly to obtain $$0.5 - \sqrt{0.249} = 0.001\ 001\ 002\ 005\ 014\ 042\ldots$$ where the decimal represen …

It is well-known that the $n$th Catalan number is equal to $(n+1)^{-1}\binom{2n}{n}$. A long time ago I had wondered what happens if you look at the sequence generated by $(n+k)^{- …

Define $[n]=(1-q^n)/(1-q)$ and $[n]!=[1][2][3] \cdots [n]$, so that $[2n]!/[n]![n+1]!$ is a polynomial in $q$ (the most algebraically natural $q$-analogue of the Catalan numbers); …

Hi guys just a quick questions What are the real life application of catalan numbers? Thanks a lot!

I believe that it is "well known" that the following two statistics on Dyck paths have symmetric joint distribution: number of returns to the axis $RET(D)$ height of the first pe …

Let $(S_n)_{n=1}^{\infty}$ be a standard random walk with $S_n = \sum_{i=1}^n X_i$ and $\mathbb{P}(X_i = \pm 1) = \frac{1}{2}$. Let $\alpha \in \mathbb{R}$ be some constant. I woul …

Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data: a noncrossing m …

I would like to ask whether there is a combinatorial proof of the following recurrence relation for Catalan numbers: $$ C_{n+1}=\frac{4n+2}{n+2} C_n. $$ Thanks!~

Given $k,n\in\mathbf{N}$ with $n\ge k$, define the set $\mathcal{F}(k,n)$ of $(k,n)$-forests of binary rooted trees (where a $(k,n)$-forest is a collection of $k$ rooted trees, whi …