Relations among partitions. I: Partitions of a finite set R. A. - - PowerPoint PPT Presentation

Relations among partitions. I: Partitions of a finite set

• R. A. Bailey

University of St Andrews Combinatorics Seminar, Shanghai Jiao Tong University, November 2017

Bailey Relations among partitions 1/24

Abstract

First I consider a single partition of a finite set. Usually it is uniform, which means that all of its parts have the same size. If it has n parts and the set has size M then we have an M × n incidence matrix and an M × M relation matrix. The partition defines an n-dimensional vector subspace

• f the M-dimensional space defined by the whole set,

as well as the matrix of orthogonal projection onto that subspace.

Bailey Relations among partitions 2/24

Abstract

First I consider a single partition of a finite set. Usually it is uniform, which means that all of its parts have the same size. If it has n parts and the set has size M then we have an M × n incidence matrix and an M × M relation matrix. The partition defines an n-dimensional vector subspace

• f the M-dimensional space defined by the whole set,

as well as the matrix of orthogonal projection onto that subspace. There is a partial order on partitions called refinement, which is related to properties of the vector subspaces and their projection matrices. This leads to the definitions of the supremum and infimum of two partitions.

Bailey Relations among partitions 2/24

Abstract

First I consider a single partition of a finite set. Usually it is uniform, which means that all of its parts have the same size. If it has n parts and the set has size M then we have an M × n incidence matrix and an M × M relation matrix. The partition defines an n-dimensional vector subspace

• f the M-dimensional space defined by the whole set,

as well as the matrix of orthogonal projection onto that subspace. There is a partial order on partitions called refinement, which is related to properties of the vector subspaces and their projection matrices. This leads to the definitions of the supremum and infimum of two partitions. Orthogonality is a nice relation between two partitions. I will give some equivalent definitions. These lead to families of mutually orthogonal partitions, such as orthogonal arrays and orthogonal block structures.

Bailey Relations among partitions 2/24

Outline

◮ One partition of a finite set. ◮ Refinement. ◮ Orthogonality.

Bailey Relations among partitions 3/24

Outline

◮ One partition of a finite set. ◮ Refinement. ◮ Orthogonality.

Bailey Relations among partitions 3/24

One partition of a finite set: the set, and some vectors

The underlying set is always denoted Ω. It is finite, of size M.

Bailey Relations among partitions 4/24

One partition of a finite set: the set, and some vectors

The underlying set is always denoted Ω. It is finite, of size M. RΩ denotes the real vector space whose coordinates are indexed by the elements of Ω. It has dimension M.

Bailey Relations among partitions 4/24

One partition of a finite set: the set, and some vectors

The underlying set is always denoted Ω. It is finite, of size M. RΩ denotes the real vector space whose coordinates are indexed by the elements of Ω. It has dimension M. V0 denotes the subspace of RΩ consisting of constant vectors. It has dimension 1.

Bailey Relations among partitions 4/24

One partition

Definition

A partition of Ω is a set of mutually disjoint non-empty subsets of Ω whose union is the whole of Ω. These subsets are called the parts of the partition.

Example

Here |Ω| = M = 12 and the partition has 3 parts, all of size 4.

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Uniform partitions

Definition

Let F be a partition of the finite set Ω. Then F is uniform (or balanced

• r homogeneous
• r proper
• r equireplicate
• r regular)

if all parts of F have the same size.

Bailey Relations among partitions 6/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F;

Bailey Relations among partitions 7/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F; ◮ if F is uniform, kF = size of each part of F;

Bailey Relations among partitions 7/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F; ◮ if F is uniform, kF = size of each part of F; ◮ VF = subspace of RΩ consisting of vectors which are

constant on each part of F;

Bailey Relations among partitions 7/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F; ◮ if F is uniform, kF = size of each part of F; ◮ VF = subspace of RΩ consisting of vectors which are

constant on each part of F;

◮ V0 ≤ VF and dim(VF) = nF;

Bailey Relations among partitions 7/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F; ◮ if F is uniform, kF = size of each part of F; ◮ VF = subspace of RΩ consisting of vectors which are

constant on each part of F;

◮ V0 ≤ VF and dim(VF) = nF; ◮ XF is the M × nF incidence matrix

• f elements of Ω in parts of F;

Bailey Relations among partitions 7/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F; ◮ if F is uniform, kF = size of each part of F; ◮ VF = subspace of RΩ consisting of vectors which are

constant on each part of F;

◮ V0 ≤ VF and dim(VF) = nF; ◮ XF is the M × nF incidence matrix

• f elements of Ω in parts of F;

◮ RF = XFX⊤ F is the M × M relation matrix for F;

Bailey Relations among partitions 7/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F; ◮ if F is uniform, kF = size of each part of F; ◮ VF = subspace of RΩ consisting of vectors which are

constant on each part of F;

◮ V0 ≤ VF and dim(VF) = nF; ◮ XF is the M × nF incidence matrix

• f elements of Ω in parts of F;

◮ RF = XFX⊤ F is the M × M relation matrix for F; ◮ PF is the matrix of orthogonal projection onto VF,

which averages each vector over each part of F.

Bailey Relations among partitions 7/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F; ◮ if F is uniform, kF = size of each part of F; ◮ VF = subspace of RΩ consisting of vectors which are

constant on each part of F;

◮ V0 ≤ VF and dim(VF) = nF; ◮ XF is the M × nF incidence matrix

• f elements of Ω in parts of F;

◮ RF = XFX⊤ F is the M × M relation matrix for F; ◮ PF is the matrix of orthogonal projection onto VF,

which averages each vector over each part of F.

Bailey Relations among partitions 7/24

Some definitions for a partition of Ω

V0 = subspace of RΩ consisting of constant vectors. For a given partition F:

◮ nF = number of parts of F; ◮ if F is uniform, kF = size of each part of F; ◮ VF = subspace of RΩ consisting of vectors which are

constant on each part of F;

◮ V0 ≤ VF and dim(VF) = nF; ◮ XF is the M × nF incidence matrix

• f elements of Ω in parts of F;

◮ RF = XFX⊤ F is the M × M relation matrix for F; ◮ PF is the matrix of orthogonal projection onto VF,

which averages each vector over each part of F. If F is uniform then PF = 1 kF XFX⊤

F = 1

kF RF.

Bailey Relations among partitions 7/24

Example with nF = 3 and kF = 4

F

Bailey Relations among partitions 8/24

Example with nF = 3 and kF = 4

F XF =                      1 1 1 1 1 1 1 1 1 1 1 1                     

Bailey Relations among partitions 8/24

Example with nF = 3 and kF = 4

F XF =                      1 1 1 1 1 1 1 1 1 1 1 1                      RF =                      1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                     

Bailey Relations among partitions 8/24

Outline

◮ One partition of a finite set. ◮ Refinement. ◮ Orthogonality.

Bailey Relations among partitions 9/24

Outline

◮ One partition of a finite set. ◮ Refinement. ◮ Orthogonality.

Bailey Relations among partitions 9/24

Refinement

Definition

Let F and G be partitions of Ω. Then F is finer than G (written F ≺ G) (and G is coarser than F) if each part of F is contained in a single part of G but nF > nG.

Bailey Relations among partitions 10/24

Refinement

Definition

Let F and G be partitions of Ω. Then F is finer than G (written F ≺ G) (and G is coarser than F) if each part of F is contained in a single part of G but nF > nG. If F ≺ G then VG < VF and so PFPG = PGPF = PG.

Bailey Relations among partitions 10/24

Refinement

Definition

Let F and G be partitions of Ω. Then F is finer than G (written F ≺ G) (and G is coarser than F) if each part of F is contained in a single part of G but nF > nG. If F ≺ G then VG < VF and so PFPG = PGPF = PG. If U is the partition with a single part (the universal partition), and E is the partition into singletons (the equality partition), then E F U for all partitions F.

Bailey Relations among partitions 10/24

Refinement

Definition

Let F and G be partitions of Ω. Then F is finer than G (written F ≺ G) (and G is coarser than F) if each part of F is contained in a single part of G but nF > nG. If F ≺ G then VG < VF and so PFPG = PGPF = PG. If U is the partition with a single part (the universal partition), and E is the partition into singletons (the equality partition), then E F U for all partitions F. VE = RΩ VU = V0 = subspace of constant vectors

Bailey Relations among partitions 10/24

Refinement

Definition

Let F and G be partitions of Ω. Then F is finer than G (written F ≺ G) (and G is coarser than F) if each part of F is contained in a single part of G but nF > nG. If F ≺ G then VG < VF and so PFPG = PGPF = PG. If U is the partition with a single part (the universal partition), and E is the partition into singletons (the equality partition), then E F U for all partitions F. VE = RΩ VU = V0 = subspace of constant vectors The relation is a partial order, which means that

◮ F F for all partitions F; ◮ if F G and G H then F H; ◮ if F G and G F then F = G.

Bailey Relations among partitions 10/24

An example with |Ω| = 64

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• Bailey

Relations among partitions 11/24

An example with |Ω| = 64

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• There are two 4-week Periods. nP = 2 and kP = 32.

Bailey Relations among partitions 11/24

An example with |Ω| = 64

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• There are two 4-week Periods. nP = 2 and kP = 32.

In each period, each homemaker does two loads of laundry (a Duo) each week. nD = 32 and kD = 2.

Bailey Relations among partitions 11/24

An example with |Ω| = 64

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• There are two 4-week Periods. nP = 2 and kP = 32.

In each period, each homemaker does two loads of laundry (a Duo) each week. nD = 32 and kD = 2. D ≺ W ≺ P and D ≺ H ≺ P.

Bailey Relations among partitions 11/24

Infimum and supremum

Definition

Let F and G be partitions of Ω. The infimum F ∧ G is the coarsest partition finer than,

• r equal to, both F and G.

Bailey Relations among partitions 12/24

Infimum and supremum

Definition

Let F and G be partitions of Ω. The infimum F ∧ G is the coarsest partition finer than,

• r equal to, both F and G.

The supremum F ∨ G is the finest partition coarser than,

• r equal to, both F and G.

Bailey Relations among partitions 12/24

Infimum and supremum

Definition

Let F and G be partitions of Ω. The infimum F ∧ G is the coarsest partition finer than,

• r equal to, both F and G.

The supremum F ∨ G is the finest partition coarser than,

• r equal to, both F and G.

So F ∧ G F F ∨ G and F ∧ G G F ∨ G.

Bailey Relations among partitions 12/24

Infimum and supremum

Definition

Let F and G be partitions of Ω. The infimum F ∧ G is the coarsest partition finer than,

• r equal to, both F and G.

The supremum F ∨ G is the finest partition coarser than,

• r equal to, both F and G.

So F ∧ G F F ∨ G and F ∧ G G F ∨ G. Every part of F ∧ G is the non-empty intersection

• f a part of F with a part of G.

Bailey Relations among partitions 12/24

Infimum and supremum

Definition

Let F and G be partitions of Ω. The infimum F ∧ G is the coarsest partition finer than,

• r equal to, both F and G.

The supremum F ∨ G is the finest partition coarser than,

• r equal to, both F and G.

So F ∧ G F F ∨ G and F ∧ G G F ∨ G. Every part of F ∧ G is the non-empty intersection

• f a part of F with a part of G.

The parts of F ∨ G are the connected components of the graph whose vertices are the elements of Ω, with an edge between ω1 and ω2 if they are in the same part of F or the same part of G.

Bailey Relations among partitions 12/24

Infimum and supremum

Definition

Let F and G be partitions of Ω. The infimum F ∧ G is the coarsest partition finer than,

• r equal to, both F and G.

The supremum F ∨ G is the finest partition coarser than,

• r equal to, both F and G.

So F ∧ G F F ∨ G and F ∧ G G F ∨ G. Every part of F ∧ G is the non-empty intersection

• f a part of F with a part of G.

The parts of F ∨ G are the connected components of the graph whose vertices are the elements of Ω, with an edge between ω1 and ω2 if they are in the same part of F or the same part of G.

Theorem

VF ∩ VG = VF∨G.

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That example with |Ω| = 64 again

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• Bailey

Relations among partitions 13/24

That example with |Ω| = 64 again

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• W ∧ H = D

and W ∨ H = P.

Bailey Relations among partitions 13/24

That example with |Ω| = 64 again

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• W ∧ H = D

and W ∨ H = P. A vector takes a constant value on each week and a constant value on each homemaker if and only if it take a constant value

• n each period, so VW ∩ VH = VP.

Bailey Relations among partitions 13/24

Outline

◮ One partition of a finite set. ◮ Refinement. ◮ Orthogonality.

Bailey Relations among partitions 14/24

Outline

◮ One partition of a finite set. ◮ Refinement. ◮ Orthogonality.

Bailey Relations among partitions 14/24

Strict orthogonality

PF = projector on VF; PG = similar; P0 = projector on V0. F(ω) = the part of F containing element ω of Ω. If F and G are partitions then VF and VG both contain V0, so these spaces cannot be orthogonal to each other.

Bailey Relations among partitions 15/24

Strict orthogonality

PF = projector on VF; PG = similar; P0 = projector on V0. F(ω) = the part of F containing element ω of Ω. If F and G are partitions then VF and VG both contain V0, so these spaces cannot be orthogonal to each other.

Definition

Partitions F and G are strictly orthogonal to each other (written F⊥G) if (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 ).

Bailey Relations among partitions 15/24

Strict orthogonality

PF = projector on VF; PG = similar; P0 = projector on V0. F(ω) = the part of F containing element ω of Ω. If F and G are partitions then VF and VG both contain V0, so these spaces cannot be orthogonal to each other.

Definition

Partitions F and G are strictly orthogonal to each other (written F⊥G) if (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 ).

Equivalent conditions Mode of thinking (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 )

angles between subspaces

Bailey Relations among partitions 15/24

Strict orthogonality

PF = projector on VF; PG = similar; P0 = projector on V0. F(ω) = the part of F containing element ω of Ω. If F and G are partitions then VF and VG both contain V0, so these spaces cannot be orthogonal to each other.

Definition

Partitions F and G are strictly orthogonal to each other (written F⊥G) if (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 ).

Equivalent conditions Mode of thinking (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 )

angles between subspaces X⊤

F (I − P0)XG = 0

matrix equation

Bailey Relations among partitions 15/24

Strict orthogonality

PF = projector on VF; PG = similar; P0 = projector on V0. F(ω) = the part of F containing element ω of Ω. If F and G are partitions then VF and VG both contain V0, so these spaces cannot be orthogonal to each other.

Definition

Partitions F and G are strictly orthogonal to each other (written F⊥G) if (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 ).

Equivalent conditions Mode of thinking (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 )

angles between subspaces X⊤

F (I − P0)XG = 0

matrix equation PFPG = PGPF = P0 matrix equation

Bailey Relations among partitions 15/24

Strict orthogonality

PF = projector on VF; PG = similar; P0 = projector on V0. F(ω) = the part of F containing element ω of Ω. If F and G are partitions then VF and VG both contain V0, so these spaces cannot be orthogonal to each other.

Definition

Partitions F and G are strictly orthogonal to each other (written F⊥G) if (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 ).

Equivalent conditions Mode of thinking (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 )

angles between subspaces X⊤

F (I − P0)XG = 0

matrix equation PFPG = PGPF = P0 matrix equation

|F∧G(ω)| |Ω|

= |F(ω)|

|Ω| × |G(ω)| |Ω|

a counting equation —proportional meeting

Bailey Relations among partitions 15/24

Some of the proofs of equivalence

The matrix of orthogonal projection onto (VF ∩ V⊥

0 ) is

PF − P0 = PF − PFP0 = PF(I − P0) = (I − P0)PF, so (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 )

⇐ ⇒ PF(I − P0)(I − P0)PG = 0 ⇐ ⇒ PF(I − P0)PG = 0 = ⇒ X⊤

F PF(I − P0)PGXG = 0

= ⇒ X⊤

F (I − P0)XG = 0,

because PGXG = XG and PFXF = XF.

Bailey Relations among partitions 16/24

Some of the proofs of equivalence

The matrix of orthogonal projection onto (VF ∩ V⊥

0 ) is

PF − P0 = PF − PFP0 = PF(I − P0) = (I − P0)PF, so (VF ∩ V⊥

0 ) ⊥ (VG ∩ V⊥ 0 )

⇐ ⇒ PF(I − P0)(I − P0)PG = 0 ⇐ ⇒ PF(I − P0)PG = 0 = ⇒ X⊤

F PF(I − P0)PGXG = 0

= ⇒ X⊤

F (I − P0)XG = 0,

because PGXG = XG and PFXF = XF. Also, PFP0 = P0 = P0PG so PF(I − P0)PG = PFPG − P0 and so PF(I − P0)PG = 0 ⇐ ⇒ PFPG = P0.

Bailey Relations among partitions 16/24

Two examples of strict orthogonality

The parts of F are rows and the parts of G are columns.

• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •

F ∧ G is uniform; so are F and G.

Bailey Relations among partitions 17/24

Two examples of strict orthogonality

The parts of F are rows and the parts of G are columns.

• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •

F ∧ G is uniform; so are F and G.

• • ••
• • •
• • •
• • • • ••
• • •
• • •

F ∧ G is not uniform; neither is F or G.

Bailey Relations among partitions 17/24

Orthogonal arrays

Definition

An orthogonal array of strength two is a collection of at least two uniform partitions on a finite set with the property that each pair is strictly orthogonal.

Bailey Relations among partitions 18/24

Orthogonal arrays

Definition

An orthogonal array of strength two is a collection of at least two uniform partitions on a finite set with the property that each pair is strictly orthogonal.

Example (11 partitions with 2 parts of size 6)

F1 1 1 1 1 1 1 F2 1 1 1 1 1 1 F3 1 1 1 1 1 1 F4 1 1 1 1 1 1 F5 1 1 1 1 1 1 F6 1 1 1 1 1 1 F7 1 1 1 1 1 1 F8 1 1 1 1 1 1 F9 1 1 1 1 1 1 F10 1 1 1 1 1 1 F11 1 1 1 1 1 1

Bailey Relations among partitions 18/24

That orthogonal array again, with each 0 replaced by −1

−1 −1 1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1

Bailey Relations among partitions 19/24

That orthogonal array again, with each 0 replaced by −1

                     −1 −1 1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1                      Add an extra row of 1s, to make a 12 × 12 matrix H.

Bailey Relations among partitions 19/24

That orthogonal array again, with each 0 replaced by −1

                     −1 −1 1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1                      Add an extra row of 1s, to make a 12 × 12 matrix H. This is a Hadamard matrix, because HH⊤ = 12I.

Bailey Relations among partitions 19/24

That orthogonal array again, with each 0 replaced by −1

                     −1 −1 1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1                      Add an extra row of 1s, to make a 12 × 12 matrix H. This is a Hadamard matrix, because HH⊤ = 12I. Orthogonal arrays, and Hadamard matrices, give rise to a substantial body of theory, with many applications.

Bailey Relations among partitions 19/24

General orthogonality

In general, VF ∩ VG = VF∨G, where F ∨ G is the supremum of F and G.

Bailey Relations among partitions 20/24

General orthogonality

In general, VF ∩ VG = VF∨G, where F ∨ G is the supremum of F and G.

Definition

Partitions F and G are orthogonal to each other (written F ⊥ G) if (VF ∩ V⊥

F∨G) ⊥ (VG ∩ V⊥ F∨G).

Bailey Relations among partitions 20/24

General orthogonality

In general, VF ∩ VG = VF∨G, where F ∨ G is the supremum of F and G.

Definition

Partitions F and G are orthogonal to each other (written F ⊥ G) if (VF ∩ V⊥

F∨G) ⊥ (VG ∩ V⊥ F∨G).

Equivalently,

◮ X⊤ F (I − PF∨G)XG = 0; ◮ PFPG = PGPF (projectors commute); ◮ proportional meeting within each part of F ∨ G.

Bailey Relations among partitions 20/24

General orthogonality

In general, VF ∩ VG = VF∨G, where F ∨ G is the supremum of F and G.

Definition

Partitions F and G are orthogonal to each other (written F ⊥ G) if (VF ∩ V⊥

F∨G) ⊥ (VG ∩ V⊥ F∨G).

Equivalently,

◮ X⊤ F (I − PF∨G)XG = 0; ◮ PFPG = PGPF (projectors commute); ◮ proportional meeting within each part of F ∨ G.

If F G then F ⊥ G.

Bailey Relations among partitions 20/24

General orthogonality

In general, VF ∩ VG = VF∨G, where F ∨ G is the supremum of F and G.

Definition

Partitions F and G are orthogonal to each other (written F ⊥ G) if (VF ∩ V⊥

F∨G) ⊥ (VG ∩ V⊥ F∨G).

Equivalently,

◮ X⊤ F (I − PF∨G)XG = 0; ◮ PFPG = PGPF (projectors commute); ◮ proportional meeting within each part of F ∨ G.

If F G then F ⊥ G. In particular, F ⊥ F for all partitions F.

Bailey Relations among partitions 20/24

Two examples of non-strict orthogonality

The parts of F are rows and the parts of G are columns.

• F ∧ G

is uniform, but F and G and F ∨ G are not.

Bailey Relations among partitions 21/24

Two examples of non-strict orthogonality

The parts of F are rows and the parts of G are columns.

• F ∧ G

is uniform, but F and G and F ∨ G are not.

• F ∧ G is not uniform;
• • ••
• neither is F or G or F ∨ G.

Bailey Relations among partitions 21/24

Another example of non-strict orthogonality

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• Bailey

Relations among partitions 22/24

Another example of non-strict orthogonality

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• Weeks are orthogonal to Homemakers but

not strictly orthogonal to Homemakers.

Bailey Relations among partitions 22/24

Another example of non-strict orthogonality

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• Weeks are orthogonal to Homemakers but

not strictly orthogonal to Homemakers. Weeks, Homemakers, Duos (W ∧ H) and Periods (W ∨ H) are all uniform.

Bailey Relations among partitions 22/24

Orthogonal block structures

U is the universal partition, with a single part. E is the equality partition, with all parts having size 1.

Definition

An orthogonal block structure is a set F of partitions of a finite set which satisfies the following.

◮ The universal partition U ∈ F. ◮ The equality partition E ∈ F. ◮ If F is in F then F is uniform. ◮ If F and G are in F then F ∨ G is in F. ◮ If F and G are in F then F ∧ G is in F. ◮ If F and G are in F then F is orthogonal to G.

Bailey Relations among partitions 23/24

An example of an orthogonal block structure

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• The first four weeks form the first Period.

The two laundry loads by Homemaker i in Week j form a Duo.

Bailey Relations among partitions 24/24

An example of an orthogonal block structure

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• The first four weeks form the first Period.

The two laundry loads by Homemaker i in Week j form a Duo. F = {E, D, W, H, P, U}.

Bailey Relations among partitions 24/24

An example of an orthogonal block structure

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• The first four weeks form the first Period.

The two laundry loads by Homemaker i in Week j form a Duo. F = {E, D, W, H, P, U}. E ≺ D ≺ W ≺ P ≺ U and D ≺ H ≺ P,

Bailey Relations among partitions 24/24

An example of an orthogonal block structure

Homemaker 1 2 3 4 5 6 7 8 1

• 2
• 3
• Week

4

• 5
• 6
• 7
• 8
• The first four weeks form the first Period.

The two laundry loads by Homemaker i in Week j form a Duo. F = {E, D, W, H, P, U}. E ≺ D ≺ W ≺ P ≺ U and D ≺ H ≺ P, W ∧ H = D, W ∨ H = P, and W is orthogonal to H.

Bailey Relations among partitions 24/24