Math.Combinat.Partitions

Partitions. Partitions are nonincreasing sequences of positive integers.

Type and basic stuff

data Partition

The additional invariant enforced here is that partitions are monotone decreasing sequences of positive integers.

Instances

Eq Partition

Ord Partition

Read Partition

Show Partition

toPartition :: [Int] -> Partition

Checks whether the input is a partition.

toPartitionUnsafe :: [Int] -> Partition

Assumes that the input is decreasing.

mkPartition :: [Int] -> Partition

Sorts the input.

isPartition :: [Int] -> Bool

fromPartition :: Partition -> [Int]

height :: Partition -> Int

The first element of the sequence.

width :: Partition -> Int

The length of the sequence.

heightWidth :: Partition -> (Int, Int)

weight :: Partition -> Int

The weight of the partition (that is, the sum of the corresponding sequence).

_dualPartition :: [Int] -> [Int]

dualPartition :: Partition -> Partition

The dual (or conjugate) partition.

Generation

_partitions'

Arguments

:: (Int, Int)

(height,width)

-> Int

d

-> [[Int]]

Partitions of d, fitting into a given rectangle, as lists.

partitions'

Arguments

:: (Int, Int)

(height,width)

-> Int

d

-> [Partition]

Partitions of d, fitting into a given rectangle. The order is again lexicographic.

countPartitions' :: (Int, Int) -> Int -> Integer

_partitions :: Int -> [[Int]]

Partitions of d, as lists

partitions :: Int -> [Partition]

Partitions of d.

countPartitions :: Int -> Integer

allPartitions'

Arguments

:: (Int, Int)

(height,width)

-> [[Partition]]

All partitions fitting into a given rectangle.

allPartitions :: Int -> [[Partition]]

All partitions up to a given degree.

countAllPartitions' :: (Int, Int) -> Integer

# = \binom { h+w } { h }

countAllPartitions :: Int -> Integer