## Mathematics, Department of

## Date of this Version

March 1959

## Abstract

Given *v* objects 1, 2, .. , *v*, a relation satisfying the following conditions is said to be an association scheme with *m* classes:

(a) Any two objects are either 1st, or 2nd, . . ,or *m*th associates, the relation of association being symmetrical, i.e., if the object α is the *i*th associate of the object β, then β is the *i*th associate of α.

(b) Each object a has *n _{i} i*th associates, the number

*n*being independent of α.

_{i}(c) If any two objects α and β are

*i*th associates, then the number of objects which are

*j*th associates of α, and

*k*th associates of β, is

*p*and is independent of the pair of

^{i}_{jk}*i*th associates α and β.

The numbers

*v, n*(

_{i}*i*= 1, 2, . ,

*m*) and

*p*(

^{i}_{jk}*i, j, k*= 1, 2, . . . ,

*m*) are the parameters of the association scheme.

If we have an association scheme with

*m*classes and given parameters, then we get a partially balanced design with

*r*replications and

*b*blocks if we can arrange the

*v*objects into

*b*sets (each set corresponding to a block) such that

(i) each set contains

*k*objects (all different) ;

(ii) each object is contained in

*r*sets;

(iii) if two objects α and β are

*i*th associates, then they occur together in

*λ*sets, the number

_{i}*λ*being independent of the particular pair of

_{i}*i*th associates α and β.

## Comments

Published in

The Annals of Mathematical Statistics, Vol. 30, No. 1. (Mar., 1959), pp. 21-38. Copyright © 1959 Institute of Mathematical Statistics. Used by permission.The Annals of Mathematical Statisticsis superceded byThe Annals of Statistics, online at http://www.imstat.org/aos/