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

In a \(\lambda\)-design \(D\), the points \(1, 2, \ldots, n\) are divided into two classes with replications \(r_1\) and \(r_2\), respectively. For any \(1 \leq i, j \leq n\), 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 \(i, j\) (\(i \neq j\)) in the same class, \(r_{ij}\) depends only on the class.