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Questions tagged [catalan-numbers]

For questions on Catalan numbers, a sequence of natural numbers that occur in various counting problems.

In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the Belgian mathematician Eugène Charles Catalan (1814–1894).

The $n$th Catalan number is given directly in terms of binomial coefficients by $$ C_n = \frac{1}{n+1}{2n\choose n} = \frac{(2n)!}{(n+1)!\,n!} = \prod\limits_{k=2}^{n}\frac{n+k}{k} \qquad\mbox{ for }n\ge 0. $$

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How do the Catalan numbers turn up here?

The Catalan numbers have a reputation for turning up everywhere, but the occurrence described below, in the analysis of an (incorrect) algorithm, is still mysterious to me, and I'm curious to find an explanation. For situations where a…
ShreevatsaR
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A conjecture about Catalan sequence

For $n=2$, consider the free abelian group generated by polynomials corresponding to $\frac{(2 n)!}{2^n n!}=3$ partitions of $2n=4$ vertices into pairs (chords): $$ (x_1-x_3)(x_2-x_4),(x_1-x_2)(x_3-x_4),(x_1-x_4)(x_2-x_3) $$ Euler's Identity:…
hbghlyj
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Simplifying Catalan number recurrence relation

While solving a problem, I reduced it in the form of the following recurrence relation. $ C_{0} = 1, C_{n} = \displaystyle\sum_{i=0}^{n - 1} C_{i}C_{n - i - 1} $ However https://en.wikipedia.org/wiki/Catalan_number tells me, this is the…
Shashwat
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Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

What is the (simple) closed form for $\large \displaystyle S(m) = \sum_{n=1}^\infty \dfrac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$? Notation: $ \dbinom{2n}n $ denotes the central binomial coefficient, $ \dfrac{(2n)!}{(n!)^2} $. We have the…
GohP.iHan
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Identity with Catalan numbers

How would you prove the following identity $$\sum_{1\ \leq\ j\ <\ j'\ \leq\ n}\ \prod_{k\ \neq\ j,\,j'}^{n} {\left(\, j + j'\,\right)^{2} \over \left(\, j - k\,\right)\left(\, j' - k\,\right)} =C_{n - 2} $$ where $C_{k}$ is the $k$-th Catalan…
abenassen
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Prove a combinatorial identity: $ \sum_{n_1+\dots+n_m=n} \prod_{i=1}^m \frac{1}{n_i}\binom{2n_i}{n_i-1}=\frac{m}{n}\binom{2n}{n-m}$

Prove the combinatorial identity $$ \sum_{n_1+\ldots+n_m=n} \;\; \prod_{i=1}^m \frac{1}{n_i}\binom{2n_i}{n_i-1}=\frac{m}{n}\binom{2n}{n-m}, \enspace n_i>0,i=1,\ldots,m $$ I "discovered" this equality during experiments with Maple, but I have…
Michael Galuza
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Catalan numbers. Sequence of balanced parentheses.

A legal sequence of parentheses is one in which the parentheses can be properly matched (each opening parenthesis should be matched to a closing one that lies further to its right). For instance, $()(())$ is a legal sequence of parentheses. I should…
stackoverload
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Show that that $\lim_{n\to\infty}\sqrt[n]{\binom{2n}{n}} = 4$

I know that $$ \lim_{n\to\infty}{{2n}\choose{n}}^\frac{1}{n} = 4 $$ but I have no Idea how to show that; I think it has something to do with reducing ${n}!$ to $n^n$ in the limit, but don't know how to get there. How might I prove that the limit is…
aaazalea
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Sum of sign of all permutations in the set of 123-avoiding permutations.

Denote $S_n$ by the symmetric group of order $n$. For a permutation $\sigma\in S_n$ we define the sign of $\sigma$ as $\def\sgn{\operatorname{sgn}} \sgn(\sigma)=1$ if it is an even permutation and $\sgn(\sigma)=-1$ if it is an odd permutation. Now,…
user1992
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Harmonic Numbers series I

Can it be shown that \begin{align} \sum_{n=1}^{\infty} \binom{2n}{n} \ \frac{H_{n+1}}{n+1} \ \left(\frac{3}{16}\right)^{n} = \frac{5}{3} + \frac{8}{3} \ \ln 2 - \frac{8}{3} \ \ln 3 \end{align} where $H_{n}$ is the Harmonic number and defined…
Leucippus
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Catalan numbers and triangulations

The number of ways to parenthesize an $n$ fold product is a Catalan number in the list $1,1,2,5,14,\cdots$ where these are in order of the number of terms in the product. The $n$th such number is also the numbr of ways to triangulate an…
coffeemath
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Identity with Harmonic and Catalan numbers

Can anyone help me with this. Prove that $$2\log \left(\sum_{n=0}^{\infty}\binom{2n}{n}\frac{x^n}{n+1}\right)=\sum_{n=1}^{\infty}\binom{2n}{n}\left(H_{2n-1}-H_n\right)\frac{x^n}{n}$$ Where $H_n=\sum_{k=1}^{n}\frac{1}{k}$. The left side is equal to…
Abou Salah
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Proving this identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$ using lattice paths

How can I prove the identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$? I have to prove it using lattice paths, it should be related to Catalan numbers The $n$th Catalan number $C_n$ counts the number of monotonic…
Mec
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Intuitive Explanation for Number of Dyck Paths Never Going Above Diagonal of a Rectangle

Suppose we have a an $a\times{}b$ rectangle whose bottom-left corner is at $(0,0)$ and whose upper-right corner is at $(b,a)$. Let $a$ and $b$ both be positive integers, and let $b\geq{}a$. If $a$ and $b$ are mutually prime then the number of Dyck…
The Riddler
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Number of ways to pair off $2n$ points such that no chords intersect

For $n \geq 0$ evenly distribute $2n$ points on the circumference of a circle, and label these point cyclically with the numbers $1, 2 . . . , 2n$ Let $h_n$ be the number of ways in which these $2n$ points can be paired off as $n$ chords where no…
alkabary
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