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Results tagged with catalan-numbers
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user 21291
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
8
votes
Intuition behind Hook Length Formula
Here is a bijection to Dyck paths (or to well-formed bracketings):
Take a SYT of shape $2\times n$ (so it contains the numbers $\{1,\ldots,2n\}$, and we aim to form a word of $n$ up-steps (opening br …
- 3,271
1
vote
Equidistribution of returns and height of first peak of Dyck paths
For future references: as we have discussed in this question, this also follows from the "zeta map" sending the bistatistic (area,bounce) to the bistatistic (dinv,area). For another definition and fur …
- 3,271
16
votes
Accepted
A double grading of catalan numbers
It appears that the zeta map which I used to answer Vince Vatter's initial question and which I describe in my answer there, see also page 50 of Jim Haglund's book, indeed solves also this problem:
A …
- 3,271
15
votes
Accepted
A family of words counted by the Catalan numbers
Below my modified answer containing a complete bijection between the above sequences and Dyck paths:
Let $a = (a_1,\ldots,a_n)$ be a sequence of $n$ integers. $a$ satisfies Property $A$ if it satisfi …
- 3,271
10
votes
Accepted
What does the $q$-Catalan Numbers count?
As Vasu commented already: there is not "the" q-analogue of the Catalan numbers. And indeed, you're mixing two different here.
Your first q-Catalan numbers defined by the $q$-binomials is MacMahon's …
- 3,271
5
votes
enumerative meaning of natural q-Catalan numbers
Another less known interpretation of MacMahon's $q$-Catalan numbers is
$$\sum_{\pi \in \mathcal{S}_n(231)} q^{\operatorname{maj}(\pi) + \operatorname{imaj}(\pi)} = \frac{1}{[n+1]_q} \begin{bmatrix} 2 …
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