Outline of combinatorics

Short description: Overview of and topical guide to combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.

Essence of combinatorics

• Matroid
• Greedoid
• Ramsey theory
  • Van der Waerden's theorem
  • Hales–Jewett theorem
  • Umbral calculus, binomial type polynomial sequences
• Combinatorial species

Branches of combinatorics

• Algebraic combinatorics
• Analytic combinatorics
• Arithmetic combinatorics
• Combinatorics on words
• Combinatorial design theory
• Enumerative combinatorics
• Extremal combinatorics
• Geometric combinatorics
• Graph theory
• Infinitary combinatorics
• Matroid theory
• Order theory
• Partition theory
• Probabilistic combinatorics
• Topological combinatorics

Multi-disciplinary fields that include combinatorics

• Coding theory
• Combinatorial optimization
• Combinatorics and dynamical systems
• Combinatorics and physics
• Discrete geometry
• Finite geometry
• Phylogenetics

History of combinatorics

History of combinatorics

General combinatorial principles and methods

• Combinatorial principles
• Trial and error, brute-force search, bogosort, British Museum algorithm
• Pigeonhole principle
• Method of distinguished element
• Mathematical induction
• Recurrence relation, telescoping series
• Generating functions as an application of formal power series
  • Cyclic sieving
  • Schrödinger method
  • Exponential generating function
  • Stanley's reciprocity theorem
• Binomial coefficients and their properties
• Combinatorial proof
  • Double counting (proof technique)
  • Bijective proof
• Inclusion–exclusion principle
• Möbius inversion formula
• Parity, even and odd permutations
• Combinatorial Nullstellensatz
• Incidence algebra
• Greedy algorithm
• Divide and conquer algorithm
  • Akra–Bazzi method
• Dynamic programming
• Branch and bound
• Birthday attack, birthday paradox
• Floyd's cycle-finding algorithm
• Reduction to linear algebra
• Sparsity
• Weight function
• Minimax algorithm
  • Alpha–beta pruning
• Probabilistic method
• Sieve methods
• Analytic combinatorics
• Symbolic combinatorics
• Combinatorial class
• Exponential formula
• Twelvefold way
• MacMahon Master theorem

Data structure concepts

• Data structure
  • Data type
  • Abstract data type
  • Algebraic data type
  • Composite type
• Array
• Associative array
• Deque
• List
  • Linked list
• Queue
  • Priority queue
• Skip list
• Stack
• Tree data structure
• Automatic garbage collection

Problem solving as an art

• Heuristic
• Inductive reasoning
• How to Solve It
• Creative problem solving
• Morphological analysis (problem-solving)

Living with large numbers

• Names of large numbers, long scale
• History of large numbers
• Graham's number
• Moser's number
• Skewes' number
• Large number notations
  • Conway chained arrow notation
  • Hyper4
  • Knuth's up-arrow notation
  • Moser polygon notation
  • Steinhaus polygon notation
• Large number effects
  • Exponential growth
  • Combinatorial explosion
  • Branching factor
  • Granularity
  • Curse of dimensionality
  • Concentration of measure

Persons influential in the field of combinatorics

• Noga Alon
• George Andrews
• József Beck
• Eric Temple Bell
• Claude Berge
• Béla Bollobás
• Peter Cameron
• Louis Comtet
• John Horton Conway
  • On Numbers and Games
  • Winning Ways for your Mathematical Plays
• Persi Diaconis
• Ada Dietz
• Paul Erdős
  • Erdős conjecture
• Philippe Flajolet
• Solomon Golomb
• Ron Graham
• Ben Green
• Tim Gowers
• Jeff Kahn
• Gil Kalai
• Gyula O. H. Katona
• Daniel J. Kleitman
• Imre Leader
• László Lovász
• Fedor Petrov
• George Pólya
• Vojtěch Rödl
• Gian-Carlo Rota
• Cecil C. Rousseau
• H. J. Ryser
• Dick Schelp
• Vera T. Sós
• Joel Spencer
• Emanuel Sperner
• Richard P. Stanley
• Benny Sudakov
• Endre Szemerédi
• Terence Tao
• Carsten Thomassen
• Jacques Touchard
• Pál Turán
• Bartel Leendert van der Waerden
• Herbert Wilf
• Richard Wilson
• Doron Zeilberger

Combinatorics scholars

• Category:Combinatorialists

Journals

• Advances in Combinatorics
• Annals of Combinatorics
• Ars Combinatoria
• Australasian Journal of Combinatorics
• Bulletin of the Institute of Combinatorics and Its Applications
• Combinatorica
• Combinatorics, Probability and Computing
• Computational Complexity
• Designs, Codes and Cryptography
• Discrete Analysis
• Discrete & Computational Geometry
• Discrete Applied Mathematics
• Discrete Mathematics
• Discrete Mathematics & Theoretical Computer Science
• Discrete Optimization
• Discussiones Mathematicae Graph Theory
• Electronic Journal of Combinatorics
• European Journal of Combinatorics
• The Fibonacci Quarterly
• Finite Fields and Their Applications
• Geombinatorics
• Graphs and Combinatorics
• Integers, Electronic Journal of Combinatorial Number Theory
• Journal of Algebraic Combinatorics
• Journal of Automata, Languages and Combinatorics
• Journal of Combinatorial Designs
• Journal of Combinatorial Mathematics and Combinatorial Computing
• Journal of Combinatorial Optimization
• Journal of Combinatorial Theory, Series A
• Journal of Combinatorial Theory, Series B
• Journal of Complexity
• Journal of Cryptology
• Journal of Graph Algorithms and Applications
• Journal of Graph Theory
• Journal of Integer Sequences (Electronic)
• Journal of Mathematical Chemistry
• Online Journal of Analytic Combinatorics
• Optimization Methods and Software
• The Ramanujan Journal
• Séminaire Lotharingien de Combinatoire
• SIAM Journal on Discrete Mathematics

Prizes

• Euler Medal
• European Prize in Combinatorics
• Fulkerson Prize
• König Prize
• Pólya Prize

See also

• List of factorial and binomial topics
• List of partition topics
• List of permutation topics
• List of puzzle topics.
• List of formal language and literal string topics

External links

• Combinatorics, a MathWorld article with many references.
• Combinatorics, from a MathPages.com portal.
• The Hyperbook of Combinatorics, a collection of math articles links.
• The Two Cultures of Mathematics by W. T. Gowers, article on problem solving vs theory building