combinat::catalan – Catalan numbers
combinat::catalan(n) returns the n-th Catalan number.
Call:
combinat::catalan(n)
Parameters:
|
n: |
nonnegative integer |
Return Value:
A positive integer.
Details:
The Catalan numbers are ubiquitous in combinatorics. For example, combinat::catalan(n) counts the Dyck words of size n, the ordered trees with n nodes, the binary trees with n+1 nodes, the complete binary trees with 2n+1 nodes, the standard tableaux with two rows of size n, the triangulations of a regular n+2-gone, or the non-crossing partitions of 
.
combinat::catalan(n) is calculated using the formula
.
We compute the 
-th Catalan number:
combinat::catalan(5)
Correct answer. But what was the question ?
We compute the first Catalan numbers:
combinat::catalan(n) $ n = 0..6
They coincide with the number of Dyck words:
combinat::dyckWords::count(n) $ n = 0..6
They also coincide with the numbers of tableaux with two rows or two columns of the same size:
1, nops(combinat::tableaux::list([2 $ n])) $ n = 1..6;
1, nops(combinat::tableaux::list([n $ 2])) $ n = 1..6;
If one uses a wrong argument, an error message is returned
combinat::catalan(-1)
Error: Wrong type of 1. argument (type 'Type::NonNegInt' expected,
got argument '-1');
during evaluation of 'combinat::catalan'