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

Abstract

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.