Math.RootLoci.Geometry.Mobius

Mobius inversion for the coarsening poset of partitions

Documentation

newtype Partition :: *

A partition of an integer. The additional invariant enforced here is that partitions are monotone decreasing sequences of positive integers. The Ord instance is lexicographical.

Constructors

Partition [Int]

Instances

Eq Partition

Methods

(==) :: Partition -> Partition -> Bool

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

Ord Partition

Methods

compare :: Partition -> Partition -> Ordering

(<) :: Partition -> Partition -> Bool

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

(>) :: Partition -> Partition -> Bool

(>=) :: Partition -> Partition -> Bool

max :: Partition -> Partition -> Partition

min :: Partition -> Partition -> Partition

Read Partition

Methods

readsPrec :: Int -> ReadS Partition

readList :: ReadS [Partition]

readPrec :: ReadPrec Partition

readListPrec :: ReadPrec [Partition]

Show Partition

Methods

showsPrec :: Int -> Partition -> ShowS

show :: Partition -> String

showList :: [Partition] -> ShowS

DrawASCII Partition

Methods

ascii :: Partition -> ASCII

CanBeEmpty Partition

Methods

isEmpty :: Partition -> Bool

empty :: Partition

HasNumberOfParts Partition

Methods

numberOfParts :: Partition -> Int

HasWidth Partition

Methods

width :: Partition -> Int

HasHeight Partition

Methods

height :: Partition -> Int

HasWeight Partition

Methods

weight :: Partition -> Int

HasDuality Partition

Methods

dual :: Partition -> Partition

Mathematica Partition

Methods

mathematica :: Partition -> String

CacheKey Partition

Methods

monoCache :: (Partition -> a) -> Partition -> a

The refinement poset of partitions

coarserThan :: Partition -> Partition -> Bool

finerThan :: Partition -> Partition -> Bool

(.==.) :: Partition -> Partition -> Bool

(./=.) :: Partition -> Partition -> Bool

(.<=.) :: Partition -> Partition -> Bool

(.>=.) :: Partition -> Partition -> Bool

(.<.) :: Partition -> Partition -> Bool

(.>.) :: Partition -> Partition -> Bool

closures

fastClosure :: Partition -> Set Partition

Fast computation of a single closure

fastAntiClosure :: Partition -> Set Partition

Fast computation of a single "anticlosure" (opposite poset)

closureSet :: Partition -> Set Partition

Caches and reuses all closures (lazily), this is the fastest version

closureSet' :: Partition -> Set Partition

The closure without the stratum itself

Mobius function

zetaOf :: Partition -> ZMod Partition

Zeta function of the refinement poset

mobiusOf :: Partition -> ZMod Partition

Mobius function of the refinement poset

helpers

firstLevelDown :: Partition -> [Partition]

Merging two parts

firstLevelUp :: Partition -> [Partition]

Splitting one part into two

set partitions

closureSetOfSetPartition :: SetPartition -> Set SetPartition

firstLevelDownSetP :: SetPartition -> [SetPartition]