Relations among partitions. II: Adjusting for one partition R. A. Bailey University of St Andrews Combinatorics Seminar, Shanghai Jiao Tong University, November 2017 Bailey Relations among partitions 1/21
Abstract Two partitions with n and m parts respectively define an n × m incidence matrix between the two partitions. Bailey Relations among partitions 2/21
Abstract Two partitions with n and m parts respectively define an n × m incidence matrix between the two partitions. I will give a few examples to show the statistical background, to motivate the idea of projecting onto the orthogonal complement of the subspace defined by a partition, which is known as adjusting for that partition. From this point of view, the usual notion of balance (in incomplete block designs) is just adjusted uniformity. I will say a little about balanced block designs. Bailey Relations among partitions 2/21
Abstract Two partitions with n and m parts respectively define an n × m incidence matrix between the two partitions. I will give a few examples to show the statistical background, to motivate the idea of projecting onto the orthogonal complement of the subspace defined by a partition, which is known as adjusting for that partition. From this point of view, the usual notion of balance (in incomplete block designs) is just adjusted uniformity. I will say a little about balanced block designs. Similarly, two partitions have adjusted orthogonality with respect to a third partition if adjusting for the third partition makes something defined by the first two really zero. Ordinary orthogonality can be recast in this light. Adjusted orthogonality can also be defined in terms of matrices, or, more combinatorially, by counting various things. Bailey Relations among partitions 2/21
Outline ◮ Incidence matrix between two partitions. ◮ Some statistical background. ◮ Balanced block designs. ◮ Adjusted orthogonality. Bailey Relations among partitions 3/21
Outline ◮ Incidence matrix between two partitions. ◮ Some statistical background. ◮ Balanced block designs. ◮ Adjusted orthogonality. Bailey Relations among partitions 3/21
Two partitions: incidence matrix The number of parts of partition F is denoted n F . X F is the M × n F matrix with entry 1 in row ω and column j if ω is in part j of F , and all other entries 0. Definition If F and G are partitions of Ω then the n F × n G incidence matrix N FG is defined by N FG = X ⊤ F X G . The entry in row i and column j is the size of the intersection of the i -th part of F with the j -th part of G . Bailey Relations among partitions 4/21
Example of an incidence matrix The parts of G are columns: n G = 12 and k G = 3. The parts of L are letters; n L = 9 and k L = 4. A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G Bailey Relations among partitions 5/21
Example of an incidence matrix The parts of G are columns: n G = 12 and k G = 3. The parts of L are letters; n L = 9 and k L = 4. A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 N LG = 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 Bailey Relations among partitions 5/21
Example of an incidence matrix The parts of G are columns: n G = 12 and k G = 3. The parts of L are letters; n L = 9 and k L = 4. A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 N LG N ⊤ N LG = 0 1 0 0 1 0 1 0 0 0 0 1 LG = 4 I + 1 ( J − I ) . 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 Bailey Relations among partitions 5/21
Outline ◮ Incidence matrix between two partitions. ◮ Some statistical background. ◮ Balanced block designs. ◮ Adjusted orthogonality. Bailey Relations among partitions 6/21
Outline ◮ Incidence matrix between two partitions. ◮ Some statistical background. ◮ Balanced block designs. ◮ Adjusted orthogonality. Bailey Relations among partitions 6/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G Bailey Relations among partitions 7/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ The underlying set has size 36 (vegetable patches). Bailey Relations among partitions 7/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ The underlying set has size 36 (vegetable patches). ◮ The partition D into districts has 4 parts of size 9. Bailey Relations among partitions 7/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ The underlying set has size 36 (vegetable patches). ◮ The partition D into districts has 4 parts of size 9. ◮ The partition G into gardens has 12 parts of size 3. Bailey Relations among partitions 7/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ The underlying set has size 36 (vegetable patches). ◮ The partition D into districts has 4 parts of size 9. ◮ The partition G into gardens has 12 parts of size 3. ◮ The partition L into letters (lettuce varieties) has 9 parts of size 4. Bailey Relations among partitions 7/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ The underlying set has size 36 (vegetable patches). ◮ The partition D into districts has 4 parts of size 9. ◮ The partition G into gardens has 12 parts of size 3. ◮ The partition L into letters (lettuce varieties) has 9 parts of size 4. Bailey Relations among partitions 7/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ The underlying set has size 36 (vegetable patches). ◮ The partition D into districts has 4 parts of size 9. ◮ The partition G into gardens has 12 parts of size 3. ◮ The partition L into letters (lettuce varieties) has 9 parts of size 4. Three binary relations: ◮ G ≺ D , G is a refinement of D ; Bailey Relations among partitions 7/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ The underlying set has size 36 (vegetable patches). ◮ The partition D into districts has 4 parts of size 9. ◮ The partition G into gardens has 12 parts of size 3. ◮ The partition L into letters (lettuce varieties) has 9 parts of size 4. Three binary relations: ◮ G ≺ D , G is a refinement of D ; ◮ L ⊥ D , L is strictly orthogonal to D ; Bailey Relations among partitions 7/21
An example of three uniform partitions of the same set A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ The underlying set has size 36 (vegetable patches). ◮ The partition D into districts has 4 parts of size 9. ◮ The partition G into gardens has 12 parts of size 3. ◮ The partition L into letters (lettuce varieties) has 9 parts of size 4. Three binary relations: ◮ G ≺ D , G is a refinement of D ; ◮ L ⊥ D , L is strictly orthogonal to D ; ◮ L ⊲ G , L is balanced with respect to G . Bailey Relations among partitions 7/21
Why orthogonal projection? Back to gardening experiment A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ 12 gardens, each containing three vegetable patches; ◮ 9 lettuce varieties, each grown on four patches. Bailey Relations among partitions 8/21
Why orthogonal projection? Back to gardening experiment A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ 12 gardens, each containing three vegetable patches; ◮ 9 lettuce varieties, each grown on four patches. Denote by Y ω the total yield of edible lettuce on patch ω . Bailey Relations among partitions 8/21
Why orthogonal projection? Back to gardening experiment A D G A B C A B C A B C B E H D E F E F D F D E C F I G H I I G H H I G ◮ 12 gardens, each containing three vegetable patches; ◮ 9 lettuce varieties, each grown on four patches. Denote by Y ω the total yield of edible lettuce on patch ω . Assume that Y ω is a random variable with expectation τ L ( ω ) + β G ( ω ) . Bailey Relations among partitions 8/21
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