Partitions. Partitions are nonincreasing sequences of positive integers.
See also Donald E. Knuth: The Art of Computer Programming, vol 4, pre-fascicle 3B.
- data Partition
- toPartition :: [Int] -> Partition
- toPartitionUnsafe :: [Int] -> Partition
- mkPartition :: [Int] -> Partition
- isPartition :: [Int] -> Bool
- fromPartition :: Partition -> [Int]
- height :: Partition -> Int
- width :: Partition -> Int
- heightWidth :: Partition -> (Int, Int)
- weight :: Partition -> Int
- dualPartition :: Partition -> Partition
- _dualPartition :: [Int] -> [Int]
- elements :: Partition -> [(Int, Int)]
- _elements :: [Int] -> [(Int, Int)]
- countAutomorphisms :: Partition -> Integer
- _countAutomorphisms :: [Int] -> Integer
- partitions' :: (Int, Int) -> Int -> [Partition]
- _partitions' :: (Int, Int) -> Int -> [[Int]]
- countPartitions' :: (Int, Int) -> Int -> Integer
- partitions :: Int -> [Partition]
- _partitions :: Int -> [[Int]]
- countPartitions :: Int -> Integer
- allPartitions' :: (Int, Int) -> [[Partition]]
- allPartitions :: Int -> [[Partition]]
- countAllPartitions' :: (Int, Int) -> Integer
- countAllPartitions :: Int -> Integer
- partitionMultiset :: (Eq a, Ord a) => [a] -> [[[a]]]
- type IntVector = UArray Int Int
- vectorPartitions :: IntVector -> [[IntVector]]
- _vectorPartitions :: [Int] -> [[[Int]]]
- fasc3B_algorithm_M :: [Int] -> [[IntVector]]
Type and basic stuff
The additional invariant enforced here is that partitions are monotone decreasing sequences of positive integers.
toPartition :: [Int] -> PartitionSource
Checks whether the input is a partition. See the note at isPartition
!
toPartitionUnsafe :: [Int] -> PartitionSource
Assumes that the input is decreasing.
mkPartition :: [Int] -> PartitionSource
Sorts the input, and cuts the nonpositive elements.
isPartition :: [Int] -> BoolSource
Note: we only check that the sequence is ordered, but we do not check for negative elements. This can be useful when working with symmetric functions. It may also change in the future...
fromPartition :: Partition -> [Int]Source
heightWidth :: Partition -> (Int, Int)Source
weight :: Partition -> IntSource
The weight of the partition (that is, the sum of the corresponding sequence).
dualPartition :: Partition -> PartitionSource
The dual (or conjugate) partition.
_dualPartition :: [Int] -> [Int]Source
elements :: Partition -> [(Int, Int)]Source
Example:
elements (toPartition [5,2,1]) == [ (1,1), (1,2), (1,3), (1,4), (1,5) , (2,1), (2,2), (2,3), (2,4) , (3,1) ]
countAutomorphisms :: Partition -> IntegerSource
Computes the number of "automorphisms" of a given partition.
_countAutomorphisms :: [Int] -> IntegerSource
Generation
Partitions of d, fitting into a given rectangle. The order is again lexicographic.
Partitions of d, fitting into a given rectangle, as lists.
partitions :: Int -> [Partition]Source
Partitions of d.
_partitions :: Int -> [[Int]]Source
Partitions of d, as lists
countPartitions :: Int -> IntegerSource
All partitions fitting into a given rectangle.
allPartitions :: Int -> [[Partition]]Source
All partitions up to a given degree.
countAllPartitions' :: (Int, Int) -> IntegerSource
# = \binom { h+w } { h }
Paritions of multisets, vector partitions
partitionMultiset :: (Eq a, Ord a) => [a] -> [[[a]]]Source
Partitions of a multiset.
vectorPartitions :: IntVector -> [[IntVector]]Source
Vector partitions. Basically a synonym for fasc3B_algorithm_M
.
_vectorPartitions :: [Int] -> [[[Int]]]Source
fasc3B_algorithm_M :: [Int] -> [[IntVector]]Source
Generates all vector partitions ("algorithm M" in Knuth). The order is decreasing lexicographic.