HaskellForMaths-0.4.9: Combinatorics, group theory, commutative algebra, non-commutative algebra

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

data Design a Source #

Constructors

 D [a] [[a]]
Instances
 Eq a => Eq (Design a) Source # Instance detailsDefined in Math.Combinatorics.Design Methods(==) :: Design a -> Design a -> Bool #(/=) :: Design a -> Design a -> Bool # Ord a => Ord (Design a) Source # Instance detailsDefined in Math.Combinatorics.Design Methodscompare :: 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) Source # Instance detailsDefined in Math.Combinatorics.Design MethodsshowsPrec :: Int -> Design a -> ShowS #show :: Design a -> String #showList :: [Design a] -> ShowS #

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 a -> [a] Source #

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

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

isDesign :: Ord a => Int -> 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 #

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

pg :: (Ord a, FinSet a, Num 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 :: (Num a2, Enum a2, Ord a1) => Design a1 -> Design a2 Source #

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

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

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

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

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

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

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

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

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

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

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

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

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

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