Math.Combinatorics.Design

A module for constructing and working with combinatorial designs.

Given integers t < k < v and lambda > 0, a t-design or t-(v,k,lambda) design is an incidence structure of points X and blocks B, where X is a set of v points, B is a collection of k-subsets of X, with the property that any t points are contained in exactly lambda blocks. If lambda = 1 and t >= 2, then a t-design is also called a Steiner system S(t,k,v).

Many designs are highly symmetric structures, having large automorphism groups. In particular, the Mathieu groups, which were the first discovered sporadic finite simple groups, turn up as the automorphism groups of the Witt designs.

Documentation

isSubset :: (Foldable t1, Foldable t2, Eq a) => t1 a -> t2 a -> Bool

data Design a

Constructors

D [a] [[a]]

Instances

Eq a => Eq (Design a)

Defined in Math.Combinatorics.Design

Methods

(==) :: Design a -> Design a -> Bool

(/=) :: Design a -> Design a -> Bool

Ord a => Ord (Design a)

Defined in Math.Combinatorics.Design

Methods

compare :: Design a -> Design a -> Ordering

(<) :: Design a -> Design a -> Bool

(<=) :: Design a -> Design a -> Bool

(>) :: Design a -> Design a -> Bool

(>=) :: Design a -> Design a -> Bool

max :: Design a -> Design a -> Design a

min :: Design a -> Design a -> Design a

Show a => Show (Design a)

Defined in Math.Combinatorics.Design

Methods

showsPrec :: Int -> Design a -> ShowS

show :: Design a -> String

showList :: [Design a] -> ShowS

design :: Ord a => ([a], [[a]]) -> Design a

toDesign :: Ord a => ([a], [[a]]) -> Design a

isValid :: Ord a => Design a -> Bool

points :: Design a -> [a]

blocks :: Design a -> [[a]]

noRepeatedBlocks :: Ord a => Design a -> Bool

tDesignParams :: Eq a => Int -> Design a -> Maybe (Int, Int, Int)

findvk :: Design a -> Maybe (Int, Int)

findlambda :: Eq a => Int -> Design a -> Maybe Int

designParams :: Eq a => Design a -> Maybe (Int, (Int, Int, Int))

isStructure :: Eq a => Int -> Design a -> Bool

isDesign :: Ord a => Int -> Design a -> Bool

is2Design :: Ord a => Design a -> Bool

isSquare :: Ord a => Design a -> Bool

incidenceMatrix :: Eq t => Design t -> [[Int]]

The incidence matrix of a design, with rows indexed by blocks and columns by points. (Note that in the literature, the opposite convention is sometimes used instead.)

subsetDesign :: (Ord a, Num a, Enum a) => a -> Int -> Design a

pairDesign :: Integral a => a -> Design a

ag2 :: (FiniteField k, Ord k) => [k] -> Design [k]

The affine plane AG(2,Fq), a 2-(q^2,q,1) design or Steiner system S(2,q,q^2).

pg2 :: (FiniteField k, Ord k) => [k] -> Design [k]

The projective plane PG(2,Fq), a square 2-(q^2+q+1,q+1,1) design or Steiner system S(2,q+1,q^2+q+1). For example, pg2 f2 is the Fano plane, a Steiner triple system S(2,3,7).

flatsDesignPG :: (Ord a, FinSet a, Num a) => Int -> [a] -> Int -> Design [a]

pg :: (Ord a, FinSet a, Num a) => Int -> [a] -> Design [a]

flatsDesignAG :: (Num a, Ord a, FinSet a) => Int -> [a] -> Int -> Design [a]

ag :: (Num a, Ord a, FinSet a) => Int -> [a] -> Design [a]

to1n :: (Num a2, Enum a2, Ord a1) => Design a1 -> Design a2

paleyDesign :: (Ord a, Num a) => [a] -> Design a

fanoPlane :: Design F7

dual :: Ord t => Design t -> Design [t]

The dual of a design

derivedDesign :: Ord t => Design t -> t -> Design t

pointResidual :: Ord t => Design t -> t -> Design t

complementaryDesign :: Ord a => Design a -> Design a

blockResidual :: Ord t => Design t -> [t] -> Design t

isDesignAut :: Ord a => Design a -> Permutation a -> Bool

incidenceGraph :: Ord a => Design a -> Graph (Either a [a])

The incidence graph of a design

designAuts :: Ord t => Design t -> [Permutation t]

Find a strong generating set for the automorphism group of a design

designAuts1 :: Ord a => Design a -> [Permutation a]

alphaL2_23 :: Permutation Integer

betaL2_23 :: Permutation Integer

gammaL2_23 :: Permutation Integer

l2_23 :: [Permutation Integer]

deltaM24 :: Permutation Integer

m24 :: [Permutation Integer]

Generators for the Mathieu group M24, a finite simple group of order 244823040

m24sgs :: [Permutation Integer]

A strong generating set for the Mathieu group M24, a finite simple group of order 244823040

m23sgs :: [Permutation Integer]

A strong generating set for the Mathieu group M23, a finite simple group of order 10200960

m22sgs :: [Permutation Integer]

A strong generating set for the Mathieu group M22, a finite simple group of order 443520

octad :: [Integer]

s_5_8_24 :: Design Integer

The Steiner system S(5,8,24), with 759 blocks, whose automorphism group is M24

s_4_7_23 :: Design Integer

The Steiner system S(4,7,23), with 253 blocks, whose automorphism group is M23

s_3_6_22 :: Design Integer

The Steiner system S(3,6,22), with 77 blocks, whose automorphism group is M22

s_5_8_24' :: Design Integer

alphaL2_11 :: Permutation Integer

betaL2_11 :: Permutation Integer

gammaL2_11 :: Permutation Integer

l2_11 :: [Permutation Integer]

deltaM12 :: Permutation Integer

hexad :: [Integer]

s_5_6_12 :: Design Integer

The Steiner system S(5,6,12), with 132 blocks, whose automorphism group is M12

s_4_5_11 :: Design Integer

The Steiner system S(4,5,11), with 66 blocks, whose automorphism group is M11

m12 :: [Permutation Integer]

Generators for the Mathieu group M12, a finite simple group of order 95040

m12sgs :: [Permutation Integer]

A strong generating set for the Mathieu group M12, a finite simple group of order 95040

m11sgs :: [Permutation Integer]

A strong generating set for the Mathieu group M11, a finite simple group of order 7920