HaskellForMaths-0.4.5: Combinatorics, group theory, commutative algebra, non-commutative algebra

Safe Haskell | None

Math.Combinatorics.Design

Description

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.

Synopsis

- isSubset :: Eq a => [a] -> [a] -> Bool
- data Design a = D [a] [[a]]
- design :: Ord a => ([a], [[a]]) -> Design a
- toDesign :: Ord a => ([a], [[a]]) -> Design a
- isValid :: Ord a => Design a -> Bool
- points :: Design t -> [t]
- blocks :: Design t -> [[t]]
- noRepeatedBlocks :: Ord t => Design t -> Bool
- tDesignParams :: Eq t => Int -> Design t -> Maybe (Int, Int, Int)
- findvk :: Design t -> Maybe (Int, Int)
- findlambda :: Eq t => Int -> Design t -> Maybe Int
- designParams :: Eq t => Design t -> Maybe (Int, (Int, Int, Int))
- isStructure :: Eq t => Int -> Design t -> Bool
- isDesign :: Ord t => Int -> Design t -> Bool
- is2Design :: Ord t => Design t -> Bool
- isSquare :: Ord a => Design a -> Bool
- incidenceMatrix :: Eq t => Design t -> [[Int]]
- subsetDesign :: (Enum a, Num a, Ord a) => a -> Int -> Design a
- pairDesign :: Integral a => a -> Design a
- ag2 :: (FiniteField k, Ord k) => [k] -> Design [k]
- pg2 :: (FiniteField k, Ord k) => [k] -> Design [k]
- flatsDesignPG :: (Num a, Ord a, FinSet a) => Int -> [a] -> Int -> Design [a]
- pg :: (Num a, Ord a, FinSet 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 :: (Enum a, Num a, Ord t) => Design t -> Design a
- paleyDesign :: (Num a, Ord a) => [a] -> Design a
- fanoPlane :: Design F7
- dual :: Ord t => Design t -> Design [t]
- 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])
- designAuts :: Ord t => Design t -> [Permutation t]
- designAuts1 :: (Ord a, Show 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]
- m24sgs :: [Permutation Integer]
- m23sgs :: [Permutation Integer]
- m22sgs :: [Permutation Integer]
- octad :: [Integer]
- s_5_8_24 :: Design Integer
- s_4_7_23 :: Design Integer
- s_3_6_22 :: Design Integer
- 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
- s_4_5_11 :: Design Integer
- m12 :: [Permutation Integer]
- m12sgs :: [Permutation Integer]
- m11sgs :: [Permutation Integer]

# Documentation

isSubset :: Eq a => [a] -> [a] -> Bool Source

data Design a Source

Constructors

D [a] [[a]] |

Instances

Eq a => Eq (Design a) |
Ord a => Ord (Design a) |
Show a => Show (Design a) |

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

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

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

points :: Design t -> [t]Source

blocks :: Design t -> [[t]]Source

noRepeatedBlocks :: Ord t => Design t -> Bool Source

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

findvk :: Design t -> Maybe (Int, Int)Source

findlambda :: Eq t => Int -> Design t -> Maybe Int Source

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

isStructure :: Eq t => Int -> Design t -> Bool Source

isDesign :: Ord t => Int -> Design t -> Bool Source

is2Design :: Ord t => Design t -> Bool Source

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

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

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

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

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

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]Source

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

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

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

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

to1n :: (Enum a, Num a, Ord t) => Design t -> Design aSource

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

fanoPlane :: Design F7 Source

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

The dual of a design

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

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

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

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

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

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

The incidence graph of a design

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

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

designAuts1 :: (Ord a, Show a) => Design a -> [Permutation a]Source

alphaL2_23 :: Permutation Integer Source

betaL2_23 :: Permutation Integer Source

gammaL2_23 :: Permutation Integer Source

l2_23 :: [Permutation Integer]Source

deltaM24 :: Permutation Integer Source

m24 :: [Permutation Integer]Source

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

m24sgs :: [Permutation Integer]Source

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

m23sgs :: [Permutation Integer]Source

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

m22sgs :: [Permutation Integer]Source

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

octad :: [Integer]Source

s_5_8_24 :: Design Integer Source

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

s_4_7_23 :: Design Integer Source

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

s_3_6_22 :: Design Integer Source

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

s_5_8_24' :: Design Integer Source

alphaL2_11 :: Permutation Integer Source

betaL2_11 :: Permutation Integer Source

gammaL2_11 :: Permutation Integer Source

l2_11 :: [Permutation Integer]Source

deltaM12 :: Permutation Integer Source

hexad :: [Integer]Source

s_5_6_12 :: Design Integer Source

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

s_4_5_11 :: Design Integer Source

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

m12 :: [Permutation Integer]Source

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

m12sgs :: [Permutation Integer]Source

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

m11sgs :: [Permutation Integer]Source

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