Combinatorics

9.

Special Counting Numbers and Sequences

9.1.

Catalan Numbers

9.1.1.

Definition and First Values

9.1.2.

Recurrence Relation

9.1.3.

Closed-Form Formula

9.1.4.

Generating Function

9.1.5.

Combinatorial Interpretations

9.1.5.1.

Balanced Parentheses

9.1.5.2.

9.1.5.3.

9.1.5.4.

Triangulations of Polygons

9.1.5.5.

Mountain Ranges

9.1.5.6.

Non-Crossing Partitions

9.1.6.

Generalizations

9.2.

Stirling Numbers

9.2.1.

Stirling Numbers of the First Kind

9.2.1.1.

Definition and Notation

9.2.1.2.

Unsigned Stirling Numbers

9.2.1.3.

Signed Stirling Numbers

9.2.1.4.

Connection to Permutation Cycles

9.2.1.5.

Recurrence Relations

9.2.1.6.

Generating Functions

9.2.2.

Stirling Numbers of the Second Kind

9.2.2.1.

Definition and Notation

9.2.2.2.

Connection to Set Partitions

9.2.2.3.

Recurrence Relations

9.2.2.4.

Generating Functions

9.2.3.

9.2.3.1.

Definition and Properties

9.2.3.2.

9.2.3.3.

Exponential Generating Function

9.2.3.4.

Dobinski's Formula

9.3.

Integer Partitions

9.3.1.

Definition and Basic Properties

9.3.2.

Partition Function p(n)

9.3.3.

Ferrers Diagrams

9.3.4.

9.3.5.

Conjugate Partitions

9.3.6.

Restricted Partitions

9.3.6.1.

Partitions into Distinct Parts

9.3.6.2.

Partitions into Odd Parts

9.3.6.3.

Partitions with Bounded Parts

9.3.7.

Generating Functions for Partitions

9.3.8.

Partition Identities

9.3.8.1.

Euler's Partition Theorem

9.3.8.2.

Pentagonal Number Theorem

9.4.

Other Important Sequences

9.4.1.

Fibonacci Numbers

9.4.2.

9.4.3.

Tribonacci Numbers

9.4.4.

9.4.5.

Motzkin Numbers

9.4.6.

Narayana Numbers

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8. Generating Functions

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10. Graph Theory and Combinatorics