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combinat::catalan

Catalan numbers

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Syntax

combinat::catalan(n)

Description

combinat::catalan(n) returns the n-th Catalan number.

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 {1, 2, …, n}.

combinat::catalan(n) is calculated using the formula

.

Examples

Example 1

We compute the first Catalan numbers:

combinat::catalan(n) $ n = 0..6

Example 2

If one uses a wrong argument, an error message is returned

combinat::catalan(-1)
Error: The object '-1' is incorrect. The type of argument number 1 must be 'Type::NonNegInt'.
  Evaluating: combinat::catalan

Parameters

n

Nonnegative integer

Return Values

Positive integer.

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