Mathematics, Department of


Date of this Version

February 1965


Published in The Annals of Mathematical Statistics, Vol. 36, No. 1. (Feb., 1965), pp. 331-336. Copyright © 1965 Institute of Mathematical Statistics. Used by permission. The Annals of Mathematical Statistics is superceded by The Annals of Statistics, online at


In an m-class partially balanced incomplete block (PBIB) design [3], any two distinct treatments are related as first, second, . . . ,or mth associates in accordance with certain rules, and the resulting classification of pairs of treatments is called an association scheme [1], [4]. Parameters, including v, ni, pijk, which depend only on the association relation between treatments and are common to all designs having a given association scheme, are called association scheme parameters. Other parameters, including b, r, k, λi, depend, in addition, on the arrangement of the treatments into blocks. Known results on two-class association scheme parameters, reviewed in this section with some changes in arrangement and notation, are used in Section 2 to prove some new relations. Dependent as they are on known necessary conditions, our theorems will not provide any new proofs of the nonexistence of particular designs. However, they are in a form which is convenient for application and are oriented toward the fundamental problem of the connection between number-theoretic properties of the parameters and combinatorial structure of the designs.

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