Relations among partitions

Abstract

Combinatorialists often consider a balanced incomplete-block design to consist of a set of points, a set of blocks, and an incidence relation between them which satisfies certain conditions. To a statistician, such a design is a set of experimental units with two partitions, one into blocks and the other into treatments; it is the relation between these two partitions which gives the design its properties. The most common binary relations between partitions that occur in statistics are refinement, orthogonality and balance. When there are more than two partitions, the binary relations may not suffice to give all the properties of the system. I shall survey work in this area, including designs such as double Youden rectangles.

Multi-part balanced incomplete-block designs

“…Remark 3 Although neither triple arrays nor 2-part 2-designs are special cases of the other, they both satisfy inequality (4). Proofs for triple arrays are in Bagchi (1998), Bailey (2017) and McSorley et al (2005), and the proof that we have given here also works for triple arrays.…”

Equitable partitions of Latin‐square graphs

“…Now, and are −3‐perfect sets if and only if this contrast is in , which happens if and only if the entries in sum to zero on each row, column and letter. This means that the partition is strictly orthogonal to each of , and (see , page 8). In the special case that , this means that forms an orthogonal array of strength two on .…”

“…Statisticians say that two partitions are orthogonal to each other if their projection matrices commute. An orthogonal block structure is defined in to be a set of pairwise‐orthogonal uniform partitions of a finite set which contains the two trivial partitions and is closed under join and meet. Thus , , and the two trivial partitions form an orthogonal block structure on .…”

Sesqui-arrays, a generalisation of triple arrays

“…The (37, 9, 2) biplane B H (9) has a unique block for which all the Hussain chains consist of three triangles. These give a 6 × 10 triple array with 15 symbols, which is shown in this form in [8,Figure 32], and a 9 × 28 triple array with 36 symbols.…”

“…For these three types of array we use the notation TA(v, k, λ rr , λ cc , λ rc : r × c), DA(v, k, λ rr , λ cc : r × c) and SA(v, k, λ rr , Γ, λ rc : r ×c) respectively, where Γ denotes the set of intersection numbers for pairs of distinct columns. The array in Figure 1, given in [31], is a TA(10, 3, 3, 2, 3 : 5 × 6); the array in Figure 2 is a DA(6, 2, 2, 1 : 3 × 4); while that in Figure 3 is a SA (8,3,4, {0, 2}, 3 : 4 × 6).…”