Some Classes of Partially Balanced Designs

Abstract

Incomplete block designs with a few replications are of practical interest to experimenters. Partially balanced incomplete block (PBIB) designs with two associate classes, and $k > r = 2$, were studied by one of the authors [1]. The present paper extends this investigation to the case $k > r$ with $\lambda_1 = 1$ and $\lambda_2 = 0$. It is shown that the parameters of all PBIB designs in this case are given by (4.27) and thus depend upon three integral parameters $k, r,$ and $t$, with the additional restrictions that \begin{equation*}\tag{i}1 \leqq t \leqq r,\end{equation*} \begin{equation*}\tag{i}rk(r - 1)(k - 1)/t(k + r - t - 1)\text{is a positive integer}.\end{equation*} For the particular case $r = 3$ it is shown that all designs with $t = 2$ or 3 necessarily exist, but if $t = 1$, then the only possible value of $k > r$ is 5. However designs with parameters (4.27) with $r = 3, t = 1$, and $k = 2$ or 3 are also combinatorially possible though they do not belong to the class $k > r$. Interesting by-products of this study are a lemma and five corollaries which give an insight into the structure of PBIB designs with $\lambda_1 = 1$ and $\lambda_2 = 0$, and $p^1_{11} = k - 2$, no special assumptions being made regarding $r$ and $k$.