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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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…
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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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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…
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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…
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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…
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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…
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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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Catalan numbers. Sequence of balanced parentheses.

A legal sequence of parentheses is one in which the parentheses can be properly matched,like ()(()). I should calculate the number of legal sequences of length $2n$, the answer is $C_n = {2n \choose n} - {2n \choose n + 1}$, how can it be proved…
stackoverload
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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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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…
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Prove that $a_n$ is a perfect square if $n$ is even without generating functions or Taylor series.

Let $a_n$ be the number of positive integers whose digits are all $1$, $3$, or $4$, and add up to $n$. For example, $a_5 = 6$, since there are six integers with the desired property: $41, 14, 311, 131, 113$, and $11111$. Prove that $a_n$ is a…
heyhuehei
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Combinatorial Interpretation of Fractional Binomial Coefficients

My question is a bit imprecise - but I hope you like it. I even strongly think it has a proper answer. The binomial coefficient $\binom{\frac{1}{2}}{n}$ is strongly related to Catalan numbers - the expression $(1-4x)^{\frac{1}{2}}$ appears when…
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