---
tag: VPAQZ0XW74
kind: concept
name: Algebraic combinatorics
msc: ["05E99", "05E05", "05E10"]
specializes: [0JKW7W7XFR]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_combinatorics
same_as:
  wikidata: Q1005603
  wikipedia: Algebraic combinatorics
---

## Summary

Algebraic combinatorics uses methods of abstract algebra to study combinatorial structures, and combinatorial methods to study algebraic objects, see the [[definition|5VDCJTZ8SW]].
Central topics are [[symmetric functions|PAJEHWXW6M]], [[Young tableaux|Q6TH9PH2R7]] and the [[representation theory of the symmetric group|1VQC05EEYW]].
Other topics are [[association schemes|C338GKB4VJ]], [[incidence algebras|1H9SFWHYEB]] of partially ordered sets, and the algebra of simplicial complexes.
Many of its results are identities that have both an algebraic proof and a bijective proof.

## Explanations

### L1 human

Algebraic combinatorics is a two-way street between counting and algebra.
In one direction, a counting problem is solved by building a group, a ring or a matrix and computing with it.
In the other direction, an algebraic quantity, such as the dimension of a representation, is computed by counting pictures.
For example, the dimensions of the irreducible representations of the symmetric group are counted by standard Young tableaux.