Chapter 10 Combinatorial Designs

Chapter 10 Combinatorial Designs

BIBD Example (a,b,c) (a,b,d) (a,c,e) (a,d,f) (a,e,f) (b,c,f) (b,d,e) (b,e,f) (c,d,e) (c,d,f) Here are 10 subsets of the 6 element set {a, b, c, d, e, f }.

BIBD Definition A balanced incomplete block design is a collection of k-subsets, called blocks, of a v-set S, k < v, such that each pair of elements of S occur together in exactly λ of the blocks. Notation (v, k, λ)-bibd or (v, k, λ) design. Alternately, we could say we have a (b, v, r, k, λ)-bibd. b is the number of blocks r is the number of times each element appears in a block

Necessary Conditions Theorem In a (v, k, λ) design with b blocks, each element occurs in r blocks such that 1 λ(v 1) = r(k 1) 2 bk = vr

Example Existence Show that no (11, 6, 2) design can exist.

Example Existence Does a (43, 7, 1)-BIBD exist?

Example Example (7, 3, 1)-BIBD

Finite Projective Plane Definition A finite projective plane is a (n 2 + n + 1, n + 1, 1) design.

Fisher s Inequality Theorem Fisher s Inequality: In any (v, k, λ) design, b v. Definition The incidence matrix of a (v, k, λ) design is a b v matrix A = (a ij ) defined by { 1 if the i a ij = th block contains the j th element 0 otherwise

Theorem We Need Theorem If A is a (v, k, λ) design, then A A = (r λ)i + λj where A is the transpose of A, I is the v v identity matrix and J is the v v matrix of all 1 s.

Proof of Fisher s Inequality Let A be the incidence matrix. We first show what A A is nonsingular by showing that its determinant is non-zero. Now, A A = r λ λ λ r λ λ λ λ.... λ λ. r Subtract the first row from each of the other rows. = r λ λ λ r r λ 0 λ r 0 r λ 0.... λ r 0. r λ

Proof of Fisher s Inequality (cont.) Now, add to the first column of the sum of all the other columns. r + (v 1)λ λ λ 0 r λ 0 = 0 0 r λ 0.... 0 0. r λ = [r + λ(v 1)](r λ) v 1 = [r + r(k 1)](r λ) v 1 = rk(r λ) v 1 But, k < v, so by property (1) r > λ, so A A 0, But, A A is a v v matrix, so the rank ρ of A A is ρ(a A) = v. Finally, since ρ(a A) ρ(a) and since ρ(a) b (A has b rows), v ρ(a) b.

Existence Example Can a (16, 6, 1) design exist?

Complementary Designs Definition Let D be a (b, v, r, k, λ) design on a set S of v elements. Then the complementary design D has as it s blocks the complements S B of the blocks B in D.

Complementary Designs Theorem Suppose that D is a (b, v, r, k, λ) design. Then D is a (b, v, b r, v k, b 2r + λ) design provided that b 2r + λ > 0. Why must b 2r + λ > 0?

Symmetric Designs Corollary If D is a symmetric (v, k, λ) design with v 2k + λ > 0 then D is a symmetric (v, v k, v 2k + λ) design.

Residual Designs Definition The (v 1, v k, r, k λ, λ) design obtained from a symmetric (v, k, λ) design by deleting all elements of one block is called a residual design. Example The residual design created from a (7, 3, 1)-BIBD (1,2,4) (2,3,5) (3,4,6) (4,5,7) (5,6,1) (6,7,2) (7,1,3)

Affine Plane Definition A design with the parameters (n 2, n, 1) is called an affine plane of order n. If we can arrange the blocks into groups so that each group contains each element exactly once, we say the design is resolvable.

Resolvability Definition A BIBD is resolvable if the blocks can be arranged into v groups so that the ( ) ( b r = v k) blocks of each group are disjoint and contain in their union each element exactly once. The groups are called resolution classes of parallel classes.

Kirkman(1850) Fifteen young ladies in a school walk out three abreast for seven days in succession; it is required to arrange them daily so that no two shall walk abreast twice.

(4,2,1) Design Suppose we wanted a league schedule for 4 teams where each team played each other team one time. How many weeks do we need? How many total games?

The Turning Trick A (2n, 2, 1) design exists for all integers n 1. But this would get tedious to develop for a large league unless we had a trick... If we wanted to construct a league schedule for 8 teams, what would the parameters of the corresponding block design be?

The Turning Trick 1 7 2 6 3 5 4

The Turning Trick 7 1 2 6 3 5 4

The Turning Trick 7 1 2 6 3 5 4

Back to the (4,2,1) Design We can go backwards from the residual design idea to build finite projective planes. These constructions show that affine planes of order n exist iff finite projective planes of order n exist. There is also a correspondence with...

Latin Squares Definition A Latin square on n symbols is an n n array such that each of the n symbols occurs exactly once in each row and in each column. The number n is called the order of the square. Example A B C D D A B C C D A B B C D A

Latin Squares and League Schedules Suppose a league schedule has been arranged for 2n teams in 2n 1 rounds. Then, define a 2n 2n array A = (a ij ) by a ii = n, a ij = k i j where the i th and j th teams play in round k. Since each team plays precisely one game per round, A is a Latin square.

Latin Squares and League Schedules Example Construct a Latin square of order 2n from a league schedule on 8 teams. 6 7 1 2 3 5 4

MOLS So just how many Latin squares are there of order n, up to labeling? Is there just one of each order? [ 1 2 2 1 Order 2 ] 1 2 3 2 3 1 3 1 2 Order 3 1 2 3 3 1 2 2 3 1

MOLS 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 Order 4 1 2 3 4 4 3 2 1 2 1 4 3 3 4 1 2

MOLS Definition Join (A, B) is the n n array where the i, j th entry is (a ij, b ij ) where a ij A and b ij B. These two Latin squares are called mutually orthogonal. For short, we say MOLS. For n 4, the MOLS are pairwise orthogonal.

MOLS Definition A complete set of MOLS of order n consists of n 1 pairwise orthogonal Latin squares. Notation: Number of MOLS of order n is given by N(n).

MOLS Definition A complete set of MOLS of order n consists of n 1 pairwise orthogonal Latin squares. Notation: Number of MOLS of order n is given by N(n). Theorem For all numbers n 3, N(n) 2, except for N(6) = 1.

MOLS Theorem N(n) 2 whenever n is odd, n 3.

MOLS Example Example Construct 2 MOLS of order 3.

MOLS Example Example Construct 2 MOLS of order 5.

Moore-MacNeish Theorem Moore-MacNeish: N(mn) min{n(m), N(n)} Corollary N(n) min{p α i i }-1

Latin Squares and Finite Projective Planes Theorem An affine plane of order n exists iff a finite projective plane of order n iff n 1 MOLS of order n exists.

Construction Example Construct an affine plane of order 4 from the three MOLS of order 4.

Initial Designs Definition A (cyclic) (v, k, λ) difference set (mod v) is a set D = {d 1, d 2,..., d k } of distinct elements of Z v such that each non-zero d Z v can be expressed in the form d = d i d j in precisely λ ways.

Back to the (7,3,1) Design Example {1, 2, 4} is a (7, 3, 1) difference set.

Justification Why is it that the design we will obtain will be balanced?

Translates Definition If D = {d 1, d 2,..., d k } is a (v, k, λ) difference set mod v) then the set D + A = {d 1 + a, d 2 + a,..., d k + A} is called a translate of D. Theorem If D = {d 1, d 2,..., d k } is a cyclic (v, k, λ) difference set then the translates D, D + 1,..., D + (v 1) are the blocks of a symmetric (v, k, λ) design.

Translate Example Example We will illustrate this with our {1, 2, 4} difference set.

Another Example Example Verify that {1, 2, 4, 10} is a (13,4,1) difference set in Z 13.

Another Example Example Verify that {1, 3, 4, 5, 9}(mod 11) yields a (11,5,2) design.

Difference Sets in Groups Other Than Z v Definition A (v, k, λ) difference set in an additive abelian group G of order v is a set D = {d 1, d 2,..., d k } of distinct elements of G such that each non-zero element g G has exactly λ representations as g = d i d j.

How The Example Gives a Design We can obtain the translates in the same manner, but how does this give us a (16,6,2) design?

Difference Systems Definition Let D 1,..., D t be sets of size k in an additive abelian group G of order v such that the differences arising from the D i give each non-zero element of G exactly λ times. The D 1,..., D t are said to form a (v, k, λ) difference system in G. Note: the D i need not be disjoint.

Difference System Example Example Show that {1, 2, 5},{1, 3, 9} form a (13,3,1) difference system in Z 13. Important: the differences are only taken within blocks.

Starters Definition A starter in an abelian group G of order 2n 1 is a set of n 1 unordered pairs {x 1, y 1 },..., {x n 1, y n 1 } of elements of G such that i. x 1, y 1,..., x n 1, y n 1 are precisely all the non-zero elements of G ii. ±(x 1 y 1 ),..., ±(x n 1 y n 1 ) are precisely the non-zero elements of G

Example of a Starter Example The pairs {1, 2}, {4, 8}, {5, 10}, {9, 7}, {3, 6} form a starter in Z 11.

Whist Tournaments Difference systems can also be used to construct whist tournaments. Definition A whist tournament, denoted Wh(4n), on 4n players is a schedule of games involving two players against two others, such that: (i) the games are arranged in 4n 1 rounds, each of n games (ii) each player plays in exactly one game each round (iii) each player partners every other player exactly once (iv) each player opposes every other player exactly twice Example Construct a Wh(4).

Example of Whist Tournament Example Verify that,0 v 4,5 1,10 v 2,8 3,7 v 6,9 is the initial round of a cyclic Wh(12).