Triple Youden Rectangles: A New Class of Fully Balanced Combinatorial Arrangements

Triple Youden rectangles are defined and examples are given. These combinatorial arrangements constitute a special class of \(k \times v\) row-and-column designs, \(k < v\), with superimposed treatments from three sets, namely a single set of \(v\) treatments and two sets of \(k\) treatments. The structure of each of these row-and-column designs incorporates that of a symmetrical balanced incomplete block design with \(v\) treatments in blocks of size \(k\). Indeed, when either of the two sets of \(k\) treatments is deleted from a \(k \times v\)  triple Youden rectangle, a \(k \times v\) double Youden rectangle is obtained; when both are deleted, a \(k \times v\) Youden square remains. The paper obtains an infinite class of triple Youden rectangles of size \(k \times (k+1)\). Then it presents a \(4 \times 13\) triple Youden rectangle which provides a balanced layout for two packs of playing-cards, and a \(7 \times 15\) triple Youden rectangle which incorporates a particularly remarkable \(7 \times 15\) Youden square. Triple Youden rectangles are fully balanced in a statistical as well as a combinatorial sense, and those discovered so far are statistically very efficient.