Math.Combinat.Partitions.Skew

Skew partitions.

Skew partitions are the difference of two integer partitions, denoted by lambda/mu.

For example

mkSkewPartition (Partition [9,7,3,2,2,1] , Partition [5,3,2,1])

creates the skew partition (9,7,3,2,2,1) / (5,3,2,1), which looks like

Basics

newtype SkewPartition

A skew partition lambda/mu is internally represented by the list [ (mu_i , lambda_i-mu_i) | i<-[1..n] ]

Constructors

SkewPartition [(Int, Int)]

Instances

Show SkewPartition

Defined in Math.Combinat.Partitions.Skew

Methods

showsPrec :: Int -> SkewPartition -> ShowS

show :: SkewPartition -> String

showList :: [SkewPartition] -> ShowS

DrawASCII SkewPartition

Defined in Math.Combinat.Partitions.Skew

Methods

ascii :: SkewPartition -> ASCII

HasDuality SkewPartition

Defined in Math.Combinat.Partitions.Skew

Methods

dual :: SkewPartition -> SkewPartition

HasWeight SkewPartition

Defined in Math.Combinat.Partitions.Skew

Methods

weight :: SkewPartition -> Int

Eq SkewPartition

Defined in Math.Combinat.Partitions.Skew

Methods

(==) :: SkewPartition -> SkewPartition -> Bool

(/=) :: SkewPartition -> SkewPartition -> Bool

Ord SkewPartition

Defined in Math.Combinat.Partitions.Skew

Methods

compare :: SkewPartition -> SkewPartition -> Ordering

(<) :: SkewPartition -> SkewPartition -> Bool

(<=) :: SkewPartition -> SkewPartition -> Bool

(>) :: SkewPartition -> SkewPartition -> Bool

(>=) :: SkewPartition -> SkewPartition -> Bool

max :: SkewPartition -> SkewPartition -> SkewPartition

min :: SkewPartition -> SkewPartition -> SkewPartition

HasShape (SkewTableau a) SkewPartition

Defined in Math.Combinat.Tableaux.Skew

Methods

shape :: SkewTableau a -> SkewPartition

mkSkewPartition :: (Partition, Partition) -> SkewPartition

mkSkewPartition (lambda,mu) creates the skew partition lambda/mu. Throws an error if mu is not a sub-partition of lambda.

safeSkewPartition :: (Partition, Partition) -> Maybe SkewPartition

Returns Nothing if mu is not a sub-partition of lambda.

skewPartitionWeight :: SkewPartition -> Int

The weight of a skew partition is the weight of the outer partition minus the the weight of the inner partition (that is, the number of boxes present).

normalizeSkewPartition :: SkewPartition -> SkewPartition

This function "cuts off" the "uninteresting parts" of a skew partition

fromSkewPartition :: SkewPartition -> (Partition, Partition)

Returns the outer and inner partition of a skew partition, respectively:

mkSkewPartition . fromSkewPartition == id

outerPartition :: SkewPartition -> Partition

The lambda part of lambda/mu

innerPartition :: SkewPartition -> Partition

The mu part of lambda/mu

dualSkewPartition :: SkewPartition -> SkewPartition

The dual skew partition (that is, the mirror image to the main diagonal)

skewPartitionElements :: SkewPartition -> [(Int, Int)]

See "partitionElements"

Listing skew partitions

skewPartitionsWithOuterShape :: Partition -> Int -> [SkewPartition]

Lists all skew partitions with the given outer shape and given (skew) weight

allSkewPartitionsWithOuterShape :: Partition -> [SkewPartition]

Lists all skew partitions with the given outer shape and any (skew) weight

skewPartitionsWithInnerShape :: Partition -> Int -> [SkewPartition]

Lists all skew partitions with the given inner shape and given (skew) weight

ASCII

asciiSkewFerrersDiagram :: SkewPartition -> ASCII

asciiSkewFerrersDiagram' :: (Char, Char) -> PartitionConvention -> SkewPartition -> ASCII