An association scheme on a finite set 
is a set partition 
, 
, ..., 
of 
. Each 
is a binary relation, meaning
a subset of 
.
The first relation is the diagonal relation 
, and the inverse 
of every
is one of the relations. Furthermore, whenever 
, the number
 |
depends only on 
,
,
and 
,
not on the particular pair 
. The constants 
are called the intersection
numbers of the scheme. An association scheme is commutative
if 
for all 
,
,
and 
.
Equivalently, an association scheme is a homogeneous coherent configuration. The adjacency matrices 
, 
, ..., 
of the relations span the Bose-Mesner
algebra of the scheme and satisfy
 |
The Bose-Mesner algebra is commutative exactly when the association scheme is commutative. A Schurian
scheme is one whose relations are the orbitals of
a transitive permutation group acting on 
. The valency of 
is the number of 
for which 
; the association scheme axioms make this number
independent of 
.
A scheme is thin if every relation has valency 1 and quasi-thin if every relation
has valency 1 or 2.
