Existence and constructions of connected block designs with given vectors of treatment replications and block sizes (Q1071445)

It is shown that, given a \(v\times 1\) vector r and a \(b\times 1\) vector k such that \(1'_ vr=1'_ bk=n\), a connected binary block design having r as its vector of treatment replications and k as its vector of block sizes exists if and only if the inequality \(n\geq v+b-1\) holds in addition to the condition of Gale and Ryser that k is majorized by the vector conjugate to r. Two general methods of constructing such designs are devised. The crucial part of the first method is a sequential procedure for transforming a disconnected binary block design into a connected binary block design, preserving in each step given vectors of treatment replications and block sizes. The crucial part of the second method is a sequential procedure for transforming a connected binary block design with the minimal number of experimental units into a connected binary block design with desired vectors of treatment replications and block sizes, preserving in each step the property of connectedness.