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 of 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 of 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 of 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 . R Ω 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 . R Ω denotes the real vector space whose coordinates are indexed by the elements of Ω . It has dimension M . V 0 denotes the subspace of R Ω 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. Bailey Relations among partitions 5/24

Uniform partitions Definition Let F be a partition of the finite set Ω . Then F is uniform (or balanced or homogeneous or proper or equireplicate or regular) if all parts of F have the same size. Bailey Relations among partitions 6/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; Bailey Relations among partitions 7/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; ◮ if F is uniform, k F = size of each part of F ; Bailey Relations among partitions 7/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; ◮ if F is uniform, k F = size of each part of F ; ◮ V F = subspace of R Ω consisting of vectors which are constant on each part of F ; Bailey Relations among partitions 7/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; ◮ if F is uniform, k F = size of each part of F ; ◮ V F = subspace of R Ω consisting of vectors which are constant on each part of F ; ◮ V 0 ≤ V F and dim ( V F ) = n F ; Bailey Relations among partitions 7/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; ◮ if F is uniform, k F = size of each part of F ; ◮ V F = subspace of R Ω consisting of vectors which are constant on each part of F ; ◮ V 0 ≤ V F and dim ( V F ) = n F ; ◮ X F is the M × n F incidence matrix of elements of Ω in parts of F ; Bailey Relations among partitions 7/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; ◮ if F is uniform, k F = size of each part of F ; ◮ V F = subspace of R Ω consisting of vectors which are constant on each part of F ; ◮ V 0 ≤ V F and dim ( V F ) = n F ; ◮ X F is the M × n F incidence matrix of elements of Ω in parts of F ; ◮ R F = X F X ⊤ F is the M × M relation matrix for F ; Bailey Relations among partitions 7/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; ◮ if F is uniform, k F = size of each part of F ; ◮ V F = subspace of R Ω consisting of vectors which are constant on each part of F ; ◮ V 0 ≤ V F and dim ( V F ) = n F ; ◮ X F is the M × n F incidence matrix of elements of Ω in parts of F ; ◮ R F = X F X ⊤ F is the M × M relation matrix for F ; ◮ P F is the matrix of orthogonal projection onto V F , which averages each vector over each part of F . Bailey Relations among partitions 7/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; ◮ if F is uniform, k F = size of each part of F ; ◮ V F = subspace of R Ω consisting of vectors which are constant on each part of F ; ◮ V 0 ≤ V F and dim ( V F ) = n F ; ◮ X F is the M × n F incidence matrix of elements of Ω in parts of F ; ◮ R F = X F X ⊤ F is the M × M relation matrix for F ; ◮ P F is the matrix of orthogonal projection onto V F , which averages each vector over each part of F . Bailey Relations among partitions 7/24

Some definitions for a partition of Ω V 0 = subspace of R Ω consisting of constant vectors. For a given partition F : ◮ n F = number of parts of F ; ◮ if F is uniform, k F = size of each part of F ; ◮ V F = subspace of R Ω consisting of vectors which are constant on each part of F ; ◮ V 0 ≤ V F and dim ( V F ) = n F ; ◮ X F is the M × n F incidence matrix of elements of Ω in parts of F ; ◮ R F = X F X ⊤ F is the M × M relation matrix for F ; ◮ P F is the matrix of orthogonal projection onto V F , which averages each vector over each part of F . If F is uniform then P F = 1 F = 1 X F X ⊤ R F . k F k F Bailey Relations among partitions 7/24

Example with n F = 3 and k F = 4 F Bailey Relations among partitions 8/24

Example with n F = 3 and k F = 4 F   1 0 0 1 0 0     1 0 0     1 0 0     0 1 0     0 1 0   X F =   0 1 0     0 1 0      0 0 1     0 0 1      0 0 1   0 0 1 Bailey Relations among partitions 8/24

Example with n F = 3 and k F = 4 F     1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0         1 0 0 1 1 1 1 0 0 0 0 0 0 0 0         1 0 0 1 1 1 1 0 0 0 0 0 0 0 0         0 1 0 0 0 0 0 1 1 1 1 0 0 0 0         0 1 0 0 0 0 0 1 1 1 1 0 0 0 0     X F = R F =     0 1 0 0 0 0 0 1 1 1 1 0 0 0 0         0 1 0 0 0 0 0 1 1 1 1 0 0 0 0          0 0 1   0 0 0 0 0 0 0 0 1 1 1 1       0 0 1   0 0 0 0 0 0 0 0 1 1 1 1          0 0 1 0 0 0 0 0 0 0 0 1 1 1 1     0 0 1 0 0 0 0 0 0 0 0 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