association scheme
Association schemes were introduced by statisticians in the 1950’s to analyze designs of statistical experiments. Today, association schemes are useful not only in experimental designs, but in other areas of mathematics such as combinatorics (coding theory) and group theory (permutation groups^{}).
There are several equivalent^{} ways to define an association schemes. Three useful ones are illustrated here:
Let $\mathrm{\Omega}$ be a nonempty set with $n$ elements, and $s$ a positive integer.
Definition 1. An association scheme $\mathcal{Q}$ on $\mathrm{\Omega}$ is a partition^{} on $\mathrm{\Omega}\times \mathrm{\Omega}$ into sets ${C}_{0},{C}_{1},\mathrm{\dots},{C}_{s}$ called associate classes, such that

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each ${C}_{i}$ is a symmetric relation^{} on $\mathrm{\Omega}$, and ${C}_{0}$ in particular is the diagonal relation,

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for $i,j,k\in \{0,1,\mathrm{\dots},s\}$, there is an integer ${p}_{ij}^{k}$ such that, for any $(a,b)\in {C}_{k}$,
$$\{c\in \mathrm{\Omega}\mid (a,c)\in {C}_{i}\text{and}(c,b)\in {C}_{j}\}={p}_{ij}^{k}.$$
If we write $C(a,b;i,j)$ for the set $\{c\in \mathrm{\Omega}\mid (a,c)\in {C}_{i}\text{and}(c,b)\in {C}_{j}\}$, then the second condition says that for any $(a,b)\in {C}_{k}$, the value $C(a,b;i,j)$ is a constant, depending only on $i,j$ and $k$, and not on the particular elements of ${C}_{k}$. This implies that, for any $i,j\in \{0,1,\mathrm{\dots},n\}$, the relation^{} ${C}_{i}\circ {C}_{j}$ is a union of (some of) the ${C}_{k}$’s.
The definition above can be restated in graph theoretic terminology. First, think of $\mathrm{\Omega}$ is a set of vertices, and twoelement subsets of $\mathrm{\Omega}$ are edges. The complete graph^{} on $\mathrm{\Omega}$ is just the set of all twoelement subsets of $\mathrm{\Omega}$. We may color the edges of the graph. Say there are colors labeled $1$ through $s$. For each color $i$, let ${C}_{i}$ be the set of edges with color $i$. Then each ${C}_{i}$ is just a symmetric relation on $\mathrm{\Omega}$, and that all the ${C}_{i}$’s, together with the diagonal relation, partition the set $\mathrm{\Omega}\times \mathrm{\Omega}$. This is basically the first condition of the definition above. In this regard, we can redefine an association scheme graph theoretically, as follows:
Definition 2. An association scheme $\mathcal{Q}$ is a surjective^{} coloring^{} on the edge set of a complete graph whose vertex set is $\mathrm{\Omega}$, by a set of $s$ colors (numbered $1$ through $s$), such that
for any $i,j,k\in \{1,\mathrm{\dots},s\}$, there is an integer ${p}_{ij}^{k}$ such that if $\mathcal{Q}(a,b)=k$ (the edge $\{a,b\}$ has color $k$), then
$$\{c\in \mathrm{\Omega}\mid \mathcal{Q}(a,c)=i\text{and}\mathcal{Q}(c,b)=j\}={p}_{ij}^{k}.$$
In words, the definition says that, for any color $k$, and any given edge $e$ with color $k$, the number of triangles (a triangle in a graph is a cycle consisting of three edges) with $e$ as an edge, and two other edges with colors $i,j$ respectively, is ${p}_{ij}^{k}$.
The first definition can also be viewed in terms of matrices, and adjacency matrices^{} more specifically. Given a finite set^{} $\mathrm{\Omega}$, a binary relation $R$ on $\mathrm{\Omega}$ naturally corresponds to matrix $A$ called the adjacency matrix of $R$. Entry $(i,j)$ is $1$ if the $i$th element and the $j$th element are related by $R$, and $0$ otherwise. If $R$ is reflexive^{}, then $A$ has all $1$’s in its diagonal, and if $R$ is symmetric^{}, then so is $A$. Also, it is easy to see that the composition^{} of two binary relations is the same as the product^{} of their corresponding adjacency matrices. Then the comment in the paragraph after the first definition is the same as saying that the adjacency matrix of ${C}_{i}\circ {C}_{j}$ is a linear combination^{} of the adjacency matrices of ${C}_{0},{C}_{1},\mathrm{\dots},{C}_{s}$. This gives us the third definition below:
Definition 3. An association scheme is a finite set $\mathcal{Q}$ of $n\times n$ nonzero matrices ${A}_{0},{A}_{1},\mathrm{\dots},{A}_{s}$ whose entries are $0$’s and $1$’s, such that

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each ${A}_{i}$ is a symmetric matrix, with ${A}_{0}={I}_{n}$, the identity matrix^{},

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${A}_{0}+{A}_{1}+\mathrm{\cdots}+{A}_{s}={J}_{n}$, the matrix whose entries are all $1$’s, and

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for any $i,j\in \{0,\mathrm{\dots},s\}$, ${A}_{i}{A}_{j}$ is a linear combination of ${A}_{0},{A}_{1},\mathrm{\dots},{A}_{s}$.
By the definitions of the matrices ${A}_{i}$ and the second condition, for every pair $(r,s)$, exactly one of the $s+1$ matrices has $1$ in cell $(r,s)$, and all others have $0$ in the corresponding cell. As a result, the $s+1$ matrices are linearly independent^{}.
Also, in view of the discussion above, it is easy to see that
$${A}_{i}{A}_{j}={p}_{ij}^{0}{A}_{0}+{p}_{ij}^{1}{A}_{1}+\mathrm{\cdots}+{p}_{ij}^{s}{A}_{s}.$$ 
Some terminology. $s$ is called the rank of the association scheme $\mathcal{Q}$. Any $a\in \mathrm{\Omega}$, an element $c\in \mathcal{Q}$ is said to be an $i$associate of $a$ if $(a,c)\in {C}_{i}$. For each $a\in \mathrm{\Omega}$, define:
$${C}_{i}(a):=\{c\in \mathrm{\Omega}\mid (a,c)\in {C}_{i}\}.$$ 
So ${C}_{i}(a)$ is the set of all $i$associates of $a$. Then
$${C}_{i}(a)\cap {C}_{j}(b)={p}_{ij}^{k},\text{whenever}(a,b)\in {C}_{k},$$ 
and because of the above equation, each ${p}_{ij}^{k}$ is called an intersection number of $\mathcal{Q}$. For each $i$, the intersection number ${p}_{ii}^{0}$ is called the valency of ${C}_{i}$, denoted by ${a}_{i}$.
Some basic properties of the intersection numbers:

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${p}_{ij}^{k}={p}_{ji}^{k}$

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$${p}_{i0}^{k}=\{\begin{array}{cc}1\hfill & \text{if}i=k\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}$$ 
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${C}_{i}(c)={p}_{ii}^{0}={a}_{i}$ for all $c\in \mathrm{\Omega}$.
References
 1 R. A. Bailey, Association Schemes, Designed Experiments, Algebra and Combinatorics, Cambridge University Press (2004)
Title  association scheme 

Canonical name  AssociationScheme 
Date of creation  20130322 19:15:03 
Last modified on  20130322 19:15:03 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 05E30 
Defines  associate class 
Defines  rank 
Defines  intersection number 
Defines  valency 