Questions tagged [catalan-numbers]
The Catalan numbers form the sequence of numbers starting 1,1,2,5,14,42,... with explicit formula $\frac{1}{n+1}\binom{2n}{n}$. It counts many combinatorial objects like planar binary trees, triangulations, noncrossing partitions, Dyck paths, etc. See https://oeis.org/A000108
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Application of Catalan number [closed]
Hi guys just a quick questions
What are the real life application of catalan numbers?
Thanks a lot!
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enumerative meaning of natural q-Catalan numbers
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); what enumerative ...
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Catalan objects associated to a univariate polynomial
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 matching on $2n$ ...
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Higher-dimensional Catalan numbers?
One could imagine defining various notions of higher-dimensional Catalan numbers,
by generalizing objects they count.
For example, because the Catalan numbers count the triangulations of convex ...
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Equidistribution of returns and height of first peak of Dyck paths
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 peak (or length of the ...
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Probability of a Random Walk crossing a straight line
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 would like to know the ...
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Modular congruences related to sums of Catalan numbers
I am curious if somebody can be helpful concerning the following
experimental observation:
There exist two rational sequences $\alpha_0,\alpha_1,\dots$ and
$\beta_0,\beta_1,\dots$, both with values ...
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Combinatorial proof of a recurrence for the Catalan numbers
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!~
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Counting $(n,k)$-forests of binary trees
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, which have a totality ...
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How does this relationship between the Catalan numbers and SU(2) generalize?
This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then.
As ...
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Is there a combinatorial reason that the (-1)st Catalan number is -1/2?
The $n$th Catalan number can be written in terms of factorials as
$$ C_n = {(2n)! \over (n+1)! n!}. $$
We can rewrite this in terms of gamma functions to define the Catalan numbers for complex $z$:
$$...
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