A Note on the Characterizations of Type-$1$ $\lambda$-Designs in Terms of $r_{ij}$

Abstract

In a $\lambda$-design $D$, the points $1,2,\dots ,n$ are divided into two classes with replications $r_1$ and $r_2$, respectively. For any $1\le i,j\le n$, let $r_{ij}$ be the number of the blocks containing $i$ and $j$. It is proven that $D$ is type-1 if and only if for any $i,j$ ($i\ne j$) in the same class, $r_{ij}$ depends only on the class.