Safe Haskell	None
Language	Haskell98

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

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 a Source

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

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 a => Int -> Design a -> Maybe (Int, Int, Int) Source

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

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

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

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

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

is2Design :: Ord a => Design a -> 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 :: (Ord a, Num a, Enum a) => a -> Int -> Design a Source

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

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

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

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

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

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

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

fanoPlane :: Design F7 Source

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

The dual of a design

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

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

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