---
tag: 5VDCJTZ8SW
kind: definition
name: Algebraic combinatorics
msc: ["05E99", "05E05", "05E10"]
defines: VPAQZ0XW74
uses: [0JKW7W7XFR, 0WCY42EV9V, PAJEHWXW6M]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_combinatorics
---

## Definition

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

1. to solve problems of [[combinatorics|0JKW7W7XFR]] with tools of abstract algebra, such as groups, [[group actions|0WCY42EV9V]], 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|0YQPZB5Q1G]] of a group acting on colourings is computed by [[Burnside's lemma|0Z47M1B2VS]] and the [[Pólya enumeration theorem|MPYQ39P01F]].
Example of aim 2: the product of two [[Schur polynomials|43VFMJCDYJ]] is a sum of Schur polynomials.
The [[Littlewood–Richardson rule|Q490XBH1Q2]] 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|3B55HNS754]].

The main objects include [[symmetric functions|PAJEHWXW6M]], [[Young tableaux|Q6TH9PH2R7]], [[association schemes|C338GKB4VJ]] and [[incidence algebras|1H9SFWHYEB]] of posets.