Math.Combinat.Partitions.NonCrossing

Non-crossing partitions.

See eg. http://en.wikipedia.org/wiki/Noncrossing_partition

Non-crossing partitions of the set [1..n] are encoded as lists of lists in standard form: Entries decreasing in each block and blocks listed in increasing order of their first entries. For example the partition in the diagram

is represented as

NonCrossing [[3],[5,4,2],[7,6,1],[9,8]]

The type of non-crossing partitions

newtype NonCrossing

A non-crossing partition of the set [1..n] in standard form: entries decreasing in each block and blocks listed in increasing order of their first entries.

Constructors

NonCrossing [[Int]]

Instances

Read NonCrossing

Defined in Math.Combinat.Partitions.NonCrossing

Methods

readsPrec :: Int -> ReadS NonCrossing

readList :: ReadS [NonCrossing]

readPrec :: ReadPrec NonCrossing

readListPrec :: ReadPrec [NonCrossing]

Show NonCrossing

Defined in Math.Combinat.Partitions.NonCrossing

Methods

showsPrec :: Int -> NonCrossing -> ShowS

show :: NonCrossing -> String

showList :: [NonCrossing] -> ShowS

HasNumberOfParts NonCrossing

Defined in Math.Combinat.Partitions.NonCrossing

Methods

numberOfParts :: NonCrossing -> Int

Eq NonCrossing

Defined in Math.Combinat.Partitions.NonCrossing

Methods

(==) :: NonCrossing -> NonCrossing -> Bool

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

Ord NonCrossing

Defined in Math.Combinat.Partitions.NonCrossing

Methods

compare :: NonCrossing -> NonCrossing -> Ordering

(<) :: NonCrossing -> NonCrossing -> Bool

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

(>) :: NonCrossing -> NonCrossing -> Bool

(>=) :: NonCrossing -> NonCrossing -> Bool

max :: NonCrossing -> NonCrossing -> NonCrossing

min :: NonCrossing -> NonCrossing -> NonCrossing

_isNonCrossing :: [[Int]] -> Bool

Checks whether a set partition is noncrossing.

Implementation method: we convert to a Dyck path and then back again, and finally compare. Probably not very efficient, but should be better than a naive check for crosses...)

_isNonCrossingUnsafe :: [[Int]] -> Bool

Warning: This function assumes the standard ordering!

_standardizeNonCrossing :: [[Int]] -> [[Int]]

Convert to standard form: entries decreasing in each block and blocks listed in increasing order of their first entries.

fromNonCrossing :: NonCrossing -> [[Int]]

toNonCrossingUnsafe :: [[Int]] -> NonCrossing

toNonCrossing :: [[Int]] -> NonCrossing

Throws an error if the input is not a non-crossing partition

toNonCrossingMaybe :: [[Int]] -> Maybe NonCrossing

setPartitionToNonCrossing :: SetPartition -> Maybe NonCrossing

If a set partition is actually non-crossing, then we can convert it

Bijection to Dyck paths

dyckPathToNonCrossingPartition :: LatticePath -> NonCrossing

Bijection between Dyck paths and noncrossing partitions

Based on: David Callan: Sets, Lists and Noncrossing Partitions

Fails if the input is not a Dyck path.

dyckPathToNonCrossingPartitionMaybe :: LatticePath -> Maybe NonCrossing

Safe version of dyckPathToNonCrossingPartition

nonCrossingPartitionToDyckPath :: NonCrossing -> LatticePath

The inverse bijection (should never fail proper NonCrossing-s)

_nonCrossingPartitionToDyckPathMaybe :: [[Int]] -> Maybe LatticePath

Safe version nonCrossingPartitionToDyckPath

Generating non-crossing partitions

nonCrossingPartitions :: Int -> [NonCrossing]

Lists all non-crossing partitions of [1..n]

Equivalent to (but orders of magnitude faster than) filtering out the non-crossing ones:

(sort $ catMaybes $ map setPartitionToNonCrossing $ setPartitions n) == sort (nonCrossingPartitions n)

nonCrossingPartitionsWithKParts

Arguments

:: Int

k = number of parts

-> Int

n = size of the set

-> [NonCrossing]

Lists all non-crossing partitions of [1..n] into k parts.

sort (nonCrossingPartitionsWithKParts k n) == sort [ p | p <- nonCrossingPartitions n , numberOfParts p == k ]

countNonCrossingPartitions :: Int -> Integer

Non-crossing partitions are counted by the Catalan numbers

countNonCrossingPartitionsWithKParts

Arguments

:: Int

k = number of parts

-> Int

n = size of the set

-> Integer

Non-crossing partitions with k parts are counted by the Naranaya numbers

randomNonCrossingPartition :: RandomGen g => Int -> g -> (NonCrossing, g)

Uniformly random non-crossing partition