| Safe Haskell | None |
|---|---|
| Language | Haskell2010 |
Math.Combinat.Partitions.Integer.Compact
Contents
Description
Compact representation of integer partitions.
Partitions are conceptually nonincreasing sequences of positive integers.
This implementation uses the compact-word-vectors library internally to provide
a much more memory-efficient Partition type that the naive lists of integer.
This is very helpful when building large tables indexed by partitions, for example;
and hopefully quite a bit faster, too.
Note: This is an internal module, you are not supposed to import it directly. It is also not fully ready to be used yet...
Synopsis
- newtype Partition = Partition WordVec
- showsPrecPartition :: Int -> Partition -> ShowS
- pattern Nil :: Partition
- pattern Cons :: Int -> Partition -> Partition
- pattern Partition_ :: [Int] -> Partition
- pattern Head :: Int -> Partition
- pattern Tail :: Partition -> Partition
- pattern Length :: Int -> Partition
- cmpLexico :: Partition -> Partition -> Ordering
- empty :: Partition
- isEmpty :: Partition -> Bool
- singleton :: Int -> Partition
- uncons :: Partition -> Maybe (Int, Partition)
- partitionTail :: Partition -> Partition
- cons :: Int -> Partition -> Partition
- snoc :: Partition -> Int -> Partition
- toExponentialForm :: Partition -> [(Int, Int)]
- fromExponentialForm :: [(Int, Int)] -> Partition
- width :: Partition -> Int
- height :: Partition -> Int
- widthHeight :: Partition -> (Int, Int)
- diffSequence :: Partition -> [Int]
- reverseDiffSequence :: Partition -> [Int]
- dualPartition :: Partition -> Partition
- toList :: Partition -> [Int]
- toDescList :: Partition -> [Int]
- toAscList :: Partition -> [Int]
- fromDescList :: [Int] -> Partition
- fromDescList' :: Int -> [Int] -> Partition
- isSubPartitionOf :: Partition -> Partition -> Bool
- dominates :: Partition -> Partition -> Bool
- pieriRule :: Partition -> Int -> [Partition]
- i2w :: Int -> Word
- w2i :: Word -> Int
- sum' :: [Word] -> Word
- safeTail :: [Int] -> [Int]
- descendToZero :: Int -> [Int]
- descendToOne :: Int -> [Int]
The compact partition data type
Instances
| Eq Partition Source # | |
| Ord Partition Source # | |
Defined in Math.Combinat.Partitions.Integer.Compact | |
| Show Partition Source # | |
Pattern synonyms
pattern Nil :: Partition Source #
Pattern sysnonyms allows us to use existing code with minimal modifications
pattern Partition_ :: [Int] -> Partition Source #
Simulated newtype constructor
Lexicographic comparison
Basic (de)constructrion
partitionTail :: Partition -> Partition Source #
partitionTail p == snd (uncons p)
snoc :: Partition -> Int -> Partition Source #
We assume that the element is not bigger than the last element!
exponential form
Width and height of the bounding rectangle
Differential sequence
diffSequence :: Partition -> [Int] Source #
From a non-increasing sequence [a1,a2,..,an] this computes the sequence of differences
[a1-a2,a2-a3,...,an-0]
reverseDiffSequence :: Partition -> [Int] Source #
From a non-increasing sequence [a1,a2,..,an] this computes the reversed sequence of differences
[ a[n]-0 , a[n-1]-a[n] , ... , a[2]-a[3] , a[1]-a[2] ]
Dual partition
dualPartition :: Partition -> Partition Source #
Conversion to list
toDescList :: Partition -> [Int] Source #
returns a descending (non-increasing) list
Conversion from list
fromDescList :: [Int] -> Partition Source #
We assume that the input is a non-increasing list of positive integers!
Partial orderings
Pieri rule
pieriRule :: Partition -> Int -> [Partition] Source #
Expands to product s[lambda]*h[k] as a sum of s[mu]-s. See https://en.wikipedia.org/wiki/Pieri's_formula
local (internally used) utility functions
descendToZero :: Int -> [Int] Source #
descendToOne :: Int -> [Int] Source #