Math 2.0

Combinatorics Definition

Algebraic combinatorics

Builds on CombinatoricsGroup actionSymmetric function

Definition

Algebraic combinatorics is the branch of mathematics with two complementary aims:

  1. to solve problems of combinatorics with tools of abstract algebra, such as groups, group actions, rings, linear algebra and representation theory;
  2. to describe algebraic objects, such as bases, structure constants, characters and dimensions, by combinatorial rules that count explicit finite structures.

Remarks

Example of aim 1: the number of orbits of a group acting on colourings is computed by Burnside's lemma and the Pólya enumeration theorem. Example of aim 2: the product of two Schur polynomials is a sum of Schur polynomials. The Littlewood–Richardson rule gives the coefficients as numbers of certain tableaux.

Non-example: a proof that a graph with a property exists, by showing that a random graph has it with positive probability, meets neither aim. It belongs to the probabilistic method.

The main objects include symmetric functions, Young tableaux, association schemes and incidence algebras of posets.

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  1. Combinatorics concept
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  4. Algebraic combinatorics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_combinatorics
About this page
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URIm2://5VDCJTZ8SW
Kinddefinition
WritingW0 stub
MSC05E99, 05E05, 05E10
DefinesAlgebraic combinatorics
UsesCombinatorics, Group action, Symmetric function
Changed2 November 2026

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