Catalan number

In the 1730s, he first established and used what was later to be known as Catalan numbers.

The Catalan numbers form the unique sequence with this property.

Taken together, these two conditions uniquely define the Catalan numbers.

He introduced the Catalan numbers to solve a combinatorial problem.

There are many counting problems in combinatorics whose solution is given by the Catalan numbers.

Bijective proofs of the formula for the Catalan numbers.

We conclude that is the Catalan number.

The sum of the rows in this triangle equal the Catalan numbers:

The numbers of ways of performing these pairings are the Catalan numbers.

The closely related Catalan numbers C are given by:

The diagonal C consists of the Catalan numbers.

It is a Catalan number.

The number of vertices in K is the nth Catalan number.

The set of all noncrossing partitions is one of many sets enumerated by the Catalan numbers.

Thus, if one of these trees is selected uniformly at random, its probability is the reciprocal of a Catalan number.

(Perhaps the most famous exercise is in volume 2, where readers are asked to prove that 66 different definitions of the Catalan numbers are equivalent.)

Davis, Tom: Catalan numbers.

The total number of different ways of writing applications of the magma operator is given by the Catalan number .

In combinatorics, the 'Catalan numbers' form a sequence of natural numbers that occur in various counting problems, often involving recursion defined objects.

The number of elements in a Tamari lattice for a sequence of n + 1 objects is the nth Catalan number.

Stack-sortable permutations may also be translated directly to and from (unlabeled) binary trees, another combinatorial class whose counting function is the sequence of Catalan numbers.

It is a generalization of the Catalan numbers, and is named after Eugène Charles Catalan.

The number of different binary trees on nodes is , the th Catalan number (assuming we view trees with identical structure as identical).

With n+1 matrices in the multiplication chain there are n binary operations and C ways of placing parenthesizes, where C is the nth Catalan number.