Combinatorial design

Symmetric arrangement of finite sets
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.

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.

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 (1850), and in practical problems, such as the scheduling of round robin tournaments (solution published 1880s). 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.

The classical core of the subject of combinatorial designs is built around balanced incomplete block designs (BIBDs), Hadamard matrices and Hadamard designs, symmetric BIBDs, Latin squares, resolvable BIBDs, difference sets, and pairwise balanced designs (PBDs). Other combinatorial designs are related to or have been developed from the study of these fundamental ones. A balanced incomplete block design or BIBD (usually called for short a block design) is a collection B of b subsets (called blocks) 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 (v,k,λ) 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 (the number of points equals the number of blocks). 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 (b ≥ v). A resolvable BIBD is a BIBD whose blocks can be partitioned into sets (called parallel classes), 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 (or any other set of n distinct symbols) 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. Two Latin squares of order n are said to be orthogonal if the set of all ordered pairs consisting of the corresponding entries in the two squares has n2 distinct members (all possible ordered pairs occur). A set of Latin squares of the same order forms a set of mutually orthogonal Latin squares (MOLS) if every pair of Latin squares in the set are orthogonal. There can be at most n − 1 squares in a set of MOLS of order n. A set of n − 1 MOLS of order n can be used to construct a projective plane of order n (and conversely). A (v, k, λ) 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 (when G is written with a multiplicative operation). If D is a difference set, and g in G, then g D = {gd: d in D} is also a difference set, and is called a translate of D. The set of all translates of a difference set D forms a symmetric BIBD. In such a design there are v elements and v blocks. Each block of the design consists of k points, each point is contained in k blocks. Any two blocks have exactly λ elements in common and any two points appear together in λ blocks. This SBIBD is called the development of D.

In particular, if λ = 1, then the difference set gives rise to a projective plane. An example of a (7,3,1) difference set in the group (an abelian group written additively) is the subset {1,2,4}. The development of this difference set gives the Fano plane. Since every difference set gives an SBIBD, the parameter set must satisfy the Bruck–Ryser–Chowla theorem, but not every SBIBD gives a difference set. 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 (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the order m > 2 then m must be a multiple of 4. Given an Hadamard matrix of order 4a in standardized form, remove the first row and first column and convert every −1 to a 0. The resulting 0–1 matrix M is the incidence matrix of a symmetric 2 − (4a − 1, 2a − 1, a − 1) design called an Hadamard 2 design. This construction is reversible, and the incidence matrix of a symmetric 2 design with these parameters can be used to form an Hadamard matrix of order 4a. When a = 2 we obtain the, by now familiar, Fano plane as an Hadamard 2 design. A pairwise balanced design (or PBD) is a set X together with a family of subsets of X (which need not have the same size and may contain repeats) such that every pair of distinct elements of X is contained in exactly λ (a positive integer) 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 (counted with multiplicity) is b.

Fisher's inequality holds for PBDs: For any non trivial PBD, v ≤ b. This result also generalizes the famous Erdős–De Bruijn theorem: For a PBD with λ = 1 having no blocks of size 1 or size v, v ≤ b, with equality if and only if the PBD is a projective plane or a near pencil.