Combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids.
Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, group testing and cryptography.
Example
Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people is in exactly one set together, every two sets have exactly one person in common, and no set contains everyone, all but one person, or exactly one person? The answer depends on n.This has a solution only if n has the form q2 + q + 1. It is less simple to prove that a solution exists if q is a prime power. It is conjectured that these are the only solutions. It has been further shown that if a solution exists for q congruent to 1 or 2 mod 4, then q is a sum of two square numbers. This last result, the Bruck–Ryser theorem, is proved by a combination of constructive methods based on finite fields and an application of quadratic forms.
When such a structure does exist, it is called a finite projective plane; thus showing how finite geometry and combinatorics intersect. When q = 2, the projective plane is called the Fano plane.
History
Combinatorial designs date to antiquity, with the Lo Shu Square being an early magic square. One of the earliest datable application of combinatorial design is found in India in the book Brhat Samhita by Varahamihira, written around 587 AD, for the purpose of making perfumes using 4 substances selected from 16 different substances using a magic square.Combinatorial designs developed along with the general growth of combinatorics from the 18th century, for example with Latin squares in the 18th century and Steiner systems in the 19th century. Designs have also been popular in recreational mathematics, such as Kirkman's schoolgirl problem, and in practical problems, such as the scheduling of round-robin tournaments. In the 20th century designs were applied to the design of experiments, notably Latin squares, finite geometry, and association schemes, yielding the field of algebraic statistics.
Fundamental combinatorial designs
The classical core of the subject of combinatorial designs is built around balanced incomplete block designs, Hadamard matrices and Hadamard designs, symmetric BIBDs, Latin squares, resolvable BIBDs, difference sets, and pairwise balanced designs. Other combinatorial designs are related to or have been developed from the study of these fundamental ones.- A balanced incomplete block design or BIBD is a collection B of b subsets of a finite set X of v elements, such that any element of X is contained in the same number r of blocks, every block has the same number k of elements, and each pair of distinct elements appear together in the same number λ of blocks. BIBDs are also known as 2-designs and are often denoted as 2- designs. As an example, when λ = 1 and b = v, we have a projective plane: X is the point set of the plane and the blocks are the lines.
- A symmetric balanced incomplete block design or SBIBD is a BIBD in which v = b. They are the single most important and well studied subclass of BIBDs. Projective planes, biplanes and Hadamard 2-designs are all SBIBDs. They are of particular interest since they are the extremal examples of Fisher's inequality.
- A resolvable BIBD is a BIBD whose blocks can be partitioned into sets, each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design. A solution of the famous 15 schoolgirl problem is a resolution of a BIBD with v = 15, k = 3 and λ = 1.
- A Latin rectangle is an r × n matrix that has the numbers 1, 2, 3, ..., n as its entries with no number occurring more than once in any row or column where r ≤ n. An n × n Latin rectangle is called a Latin square. If r < n, then it is possible to append n − r rows to an r × n Latin rectangle to form a Latin square, using Hall's marriage theorem.
- A difference set is a subset D of a group G such that the order of G is v, the size of D is k, and every nonidentity element of G can be expressed as a product d1d2−1 of elements of D in exactly λ ways.
- An Hadamard matrix of order m is an m × m matrix H whose entries are ±1 such that HH⊤ = mIm, where H⊤ is the transpose of H and Im is the m × m identity matrix. An Hadamard matrix can be put into standardized form where the first row and first column entries are all +1. If the order m > 2 then m must be a multiple of 4.
- A pairwise balanced design is a set X together with a family of subsets of X such that every pair of distinct elements of X is contained in exactly λ subsets. The set X is allowed to be one of the subsets, and if all the subsets are copies of X, the PBD is called trivial. The size of X is v and the number of subsets in the family is b.
Other combinatorial designs
- Association schemes
- A balanced ternary design BTD is an arrangement of V elements into B multisets, each of cardinality K, satisfying:
- Each element appears R = ρ1 + 2ρ2 times altogether, with multiplicity one in exactly ρ1 blocks and multiplicity two in exactly ρ2 blocks.
- Every pair of distinct elements appears Λ times ; that is, if mvb is the multiplicity of the element v in block b, then for every pair of distinct elements v and w,.
| 1 | 1 | 1 | 2 | 2 | 3 | 1 | 1 |
| 1 | 1 | 1 | 2 | 2 | 3 | 2 | 2 |
| 2 | 3 | 4 | 3 | 4 | 4 | 3 | 3 |
| 2 | 3 | 4 | 3 | 4 | 4 | 4 | 4 |
- A ' of order n is an arrangement of all the distinct unordered pairs of a 2n-set V into an n × array such that
- every element of V appears precisely once in each column, and
- every element of V appears at most twice in each row.
| 1 6 | 3 5 | 2 3 | 4 5 | 2 4 |
| 2 5 | 4 6 | 1 4 | 1 3 | 3 6 |
| 3 4 | 1 2 | 5 6 | 2 6 | 1 5 |
- Bent functions
- Costas arrays
- Factorial designs
- A frequency square is a higher order generalization of a Latin square. Let S = be a set of distinct symbols and a frequency vector of positive integers. A frequency square of order n is an n × n array in which each symbol si occurs λi times, i = 1,2,...,m, in each row and column. The order n = λ1 + λ2 + ... + λm. An F-square is in standard form if in the first row and column, all occurrences of si precede those of sj whenever i < j.
- Hall triple systems are Steiner triple systems with the property that the substructure generated by two intersecting lines is isomorphic to the finite affine plane AG.
- Let S be a set of 2n elements. A Howell design, H is an s × s array such that:
- Each cell of the array is either empty or contains an unordered pair from S,
- Each symbol occurs exactly once in each row and column of the array, and
- Every unordered pair of symbols occurs in at most one cell of the array.
| 0 4 | 1 3 | 2 5 | |
| 2 3 | 1 4 | 0 5 | |
| 3 5 | 2 4 | 0 1 | |
| 1 5 | 0 2 | 3 4 |
- Linear spaces
- An -lotto design is an n-set V of elements together with a set β of k-element subsets of V, so that for any p-subset P of V, there is a block B in β for which |P ∩ B | ≥ t. L denotes the smallest number of blocks in any -lotto design. The following is a -lotto design with the smallest possible number of blocks:
- Magic squares
- A -Mendelsohn design, or MD,is a v-set V and a collection β of ordered k-tuples of distinct elements of V, such that each ordered pair with x ≠ y of elements of V is cyclically adjacent in λ blocks. The ordered pair of distinct elements is cyclically adjacent in a block if the elements appear in the block as or. An MD is a Mendelsohn triple system, MTS. An example of an MTS on V = is:
- Orthogonal arrays
- A quasi-3 design is a symmetric design in which each triple of blocks intersect in either x or y points, for fixed x and y called the triple intersection numbers. Any symmetric design with λ ≤ 2 is a quasi-3 design with x = 0 and y = 1. The point-hyperplane design of PG is a quasi-3 design with x = / and y = λ = /. If y = λ for a quasi-3 design, the design is isomorphic to PG or a projective plane.
- A t- design D is quasi-symmetric with intersection numbers x and y if every two distinct blocks intersect in either x or y points. These designs naturally arise in the investigation of the duals of designs with λ = 1. A non-symmetric 2- design is quasisymmetric with x = 0 and y = 1. A multiple of a symmetric 2- design is quasisymmetric with x = λ and y = k. Hadamard 3-designs are quasisymmetric.
- Room squares
- A spherical design is a finite set X of points in a -dimensional sphere such that, for some integer t, the average value on X of every polynomial
- Turán systems
- An r × n tuscan-k rectangle on n symbols has r rows and n columns such that:
- each row is a permutation of the n symbols and
- for any two distinct symbols a and b and for each m from 1 to k, there is at most one row in which b is m steps to the right of a.
| 6 | 1 | 5 | 2 | 4 | 3 | 7 |
| 2 | 6 | 3 | 5 | 4 | 7 | 1 |
| 5 | 7 | 2 | 3 | 1 | 4 | 6 |
| 4 | 2 | 5 | 1 | 6 | 7 | 3 |
| 3 | 6 | 2 | 1 | 7 | 4 | 5 |
| 1 | 3 | 2 | 7 | 5 | 6 | 4 |
| 7 | 6 | 5 | 3 | 4 | 1 | 2 |
- A t-wise balanced design' of type t − is a v-set X together with a family of subsets of X whose sizes are in the set K, such that every t-subset of distinct elements of X is contained in exactly λ blocks. If K is a set of positive integers strictly between t and v, then the t BD is proper. If all the k-subsets of X for some k are blocks, the t BD is a trivial design.