combinatorics

Table of Contents

Introduction
History
- Early developments
- Combinatorics during the 20th century
Problems of enumeration
- Permutations and combinations
  - Binomial coefficients
  - Multinomial coefficients
- Recurrence relations and generating functions
- Partitions
- The Ferrer diagram
- The principle of inclusion and exclusion: derangements
- Polya’s theorem
- The Möbius inversion theorem
- Special problems
  - The Ising problem
  - Self-avoiding random walk
Problems of choice
- Systems of distinct representatives
- Ramsey’s numbers
Design theory
- BIB (balanced incomplete block) designs
- PBIB (partially balanced incomplete block) designs
Latin squares and the packing problem
- Orthogonal Latin squares
- Orthogonal arrays and the packing problem
Graph theory
- Definitions
- Enumeration of graphs
- Characterization problems of graph theory
Applications of graph theory
- Planar graphs
- The four-colour map problem
- Eulerian cycles and the Königsberg bridge problem
- Directed graphs
Combinatorial geometry
- Some historically important topics of combinatorial geometry
  - Packing and covering
  - Polytopes
  - Incidence problems
  - Helly’s theorem
- Methods of combinatorial geometry
  - Exhausting the possibilities
  - Use of extremal properties
  - Use of transformations between different spaces and applications of Helly’s theorem

References & Edit History Quick Facts & Related Topics

Images

Ferrers' partitioning diagram for 14

orthogonal Latin squares

two isomorphic graphs and a tree

two homeomorphic graphs A and B

two graphs important to planar properties

seven bridges of Königsberg and multigraph.

packing of disks

covering of part of a plane with triangles

prism and truncated pyramid

example of theorem on extremal properties

For Students

combinatorics summary

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Parsons Weems' Fable, oil on canvas by Grant Wood, 1939; in the Amon Carter Museum of American Art. Catalog entry: The historian Mason Locke Weems, commonly known by his clerical title, Parson Weems, pulls back a curtain to point at 6-year-old George Washington. Bearing an adult head and holding an axe, George gazes blankly up at his father, Augustine, who questions his son while holding a partially cut cherry tree. All of this takes place in front of a 20th-century brick building, which Wood modeled after his own home in Iowa. The scene illustrates the fable of Washington and the cherry tree, which Weems invented in 1806 for the first president's biography. Wood playfully suggests the artifice of the tale here, presenting the story in the manner of a costumed stage play rather than a real event. Amid the action, two Black figures pick cherries in the background, a reference to Washington' slaveholding past and a reminder of of what often gets left out in historical mythmaking.

Did George Washington Really Say, “I Can’t Tell a Lie”?

In this aerial photo, structures are damaged and destroyed October 15, 2005 in Balakot, Pakistan. It is estimated that 90% of the city of Balakot was leveled by the earthquake. The death toll in the 7.6 magnitude earthquake that struck northern Pakistan on October 8, 2005 is believed to be 38,000 with at least 1,300 more dead in Indian Kashmir. SEE CONTENT NOTES.

6 of the World’s Deadliest Natural Disasters

Shah Jahan. Taj Mahal. Mughal architecture. Emperor Shah Jahan fifth Mughal Emperor (reigned 1628-1658) India, Himachal Pradesh, Basohli or Jammu and Kashmir, Mankot, circa 1690 Drawings; Opaque watercolor, gold, and ink on paper (see notes)

6 Important Mughal Emperors

The Lost Colony of Roanoke, North Carolina, where 115 people mysteriously disappeared c. 1590. John White discovers the word Croatoan carved onto a tree upon his return to the deserted Roanoke Colony in 1590

The Lost Colony of Roanoke

pg 229Nazi parade features a banner proclaiming, "Death to Marxism."The possibility of a peaceful Germany after World War I was precluded entirely by the terms of the Versailles Treaty and theintransigent hostility of France and England. Stripped of indu

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Chapter 4 pg 42 - Chapter header of The Adventures of Tom Sawyer by Mark Twain. Published in 1884 by The American Publishing Company

Periods of American Literature

The Colosseum, Rome, Italy. Giant amphitheatre built in Rome under the Flavian emperors. (ancient architecture; architectural ruins)

New Seven Wonders of the World

combinatorics: Facts & Related Content

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Facts

Also Known As	combinatorial mathematics

Photos

Ferrers' partitioning diagram for 14

orthogonal Latin squares

two isomorphic graphs and a tree

two homeomorphic graphs A and B

two graphs important to planar properties

seven bridges of Königsberg and multigraph.

packing of disks

covering of part of a plane with triangles

prism and truncated pyramid

example of theorem on extremal properties

See All Images →

Topics

necklace problem graph theory permutations and combinations distinct representative Ising problem Möbius inversion theorem Ramsey’s numbers orthogonal array Latin square design statistics analysis probability theory cryptology computer science algebra geometry automata theory set theory information theory

Key People

Paul Erdős

Paul Erdős

Hungarian mathematician

Andrei Okounkov

Russian mathematician

Quiz

Equations written on blackboard

Numbers and Mathematics