The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and the theory of error-correcting codes. In algebra, the theory of association schemes generalizes the character theory of linear representations of groups.

An n-class association scheme consists of a set X together with a partition S of X × X into n + 1 binary relations, R₀, R₁, ..., R_n which satisfy:

An association scheme is commutative if ${\displaystyle p_{ij}^{k}=p_{ji}^{k}}$ for all ${\displaystyle i}$, ${\displaystyle j}$ and ${\displaystyle k}$. Most authors assume this property. Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group.

A symmetric association scheme is one in which each ${\displaystyle R_{i}}$ is a symmetric relation. That is:

Every symmetric association scheme is commutative.

Two points x and y are called i th associates if ${\displaystyle (x,y)\in R_{i}}$. The definition states that if x and y are i th associates then so are y and x. Every pair of points are i th associates for exactly one ${\displaystyle i}$. Each point is its own zeroth associate while distinct points are never zeroth associates. If x and y are k th associates then the number of points ${\displaystyle z}$ which are both i th associates of ${\displaystyle x}$ and j th associates of ${\displaystyle y}$ is a constant ${\displaystyle p_{ij}^{k}}$.

A symmetric association scheme can be visualized as a complete graph with labeled edges. The graph has ${\displaystyle v}$ vertices, one for each point of ${\displaystyle X}$, and the edge joining vertices ${\displaystyle x}$ and ${\displaystyle y}$ is labeled ${\displaystyle i}$ if ${\displaystyle x}$ and ${\displaystyle y}$ are ${\displaystyle i}$ th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled ${\displaystyle k}$ having the other edges labeled ${\displaystyle i}$ and ${\displaystyle j}$ is a constant ${\displaystyle p_{ij}^{k}}$, depending on ${\displaystyle i,j,k}$ but not on the choice of the base. In particular, each vertex is incident with exactly ${\displaystyle p_{ii}^{0}=v_{i}}$ edges labeled ${\displaystyle i}$; ${\displaystyle v_{i}}$ is the valency of the relation ${\displaystyle R_{i}}$. There are also loops labeled ${\displaystyle 0}$ at each vertex ${\displaystyle x}$, corresponding to ${\displaystyle R_{0}}$.

The relations are described by their adjacency matrices. ${\displaystyle A_{i}}$ is the adjacency matrix of ${\displaystyle R_{i}}$ for ${\displaystyle i=0,\ldots ,n}$ and is a v × v matrix with rows and columns labeled by the points of ${\displaystyle X}$.