combinat-0.2.8.1: Generate and manipulate various combinatorial objects.

Math.Combinat.Partitions.Integer

Description

Partitions of integers. Integer partitions are nonincreasing sequences of positive integers.

See:

For example the partition

Partition [8,6,3,3,1]

can be represented by the (English notation) Ferrers diagram:

Synopsis

# Type and basic stuff

newtype Partition Source

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]

mkPartition :: [Int] -> Partition Source

Sorts the input, and cuts the nonpositive elements.

Assumes that the input is decreasing.

toPartition :: [Int] -> Partition Source

Checks whether the input is an integer partition. See the note at isPartition!

isPartition :: [Int] -> Bool Source

This returns True if the input is non-increasing sequence of positive integers (possibly empty); False otherwise.

The first element of the sequence.

The length of the sequence (that is, the number of parts).

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

The dual (or conjugate) partition.

data Pair Source

Constructors

 Pair !Int !Int

_dualPartitionNaive :: [Int] -> [Int] Source

A simpler, but bit slower (about twice?) implementation of dual partition

diffSequence :: [Int] -> [Int] Source

From a sequence [a1,a2,..,an] computes the sequence of differences [a1-a2,a2-a3,...,an-0]

elements :: Partition -> [(Int, Int)] Source

Example:

elements (toPartition [5,4,1]) ==
[ (1,1), (1,2), (1,3), (1,4), (1,5)
, (2,1), (2,2), (2,3), (2,4)
, (3,1)
]

_elements :: [Int] -> [(Int, Int)] Source

# Exponential form

toExponentialForm :: Partition -> [(Int, Int)] Source

We convert a partition to exponential form. (i,e) mean (i^e); for example [(1,4),(2,3)] corresponds to (1^4)(2^3) = [2,2,2,1,1,1,1]. Another example:

toExponentialForm (Partition [5,5,3,2,2,2,2,1,1]) == [(1,2),(2,4),(3,1),(5,2)]

# Automorphisms

Computes the number of "automorphisms" of a given integer partition.

# Generating partitions

partitions :: Int -> [Partition] Source

Partitions of d.

_partitions :: Int -> [[Int]] Source

Partitions of d, as lists

Number of partitions of n

This uses countPartitions', and thus is slow

Infinite list of number of partitions of 0,1,2,...

This uses the infinite product formula the generating function of partitions, recursively expanding it; it is quite fast.

partitionCountList == map countPartitions [0..]

Naive infinite list of number of partitions of 0,1,2,...

partitionCountListNaive == map countPartitionsNaive [0..]

This is much slower than the power series expansion above.

All integer partitions up to a given degree (that is, all integer partitions whose sum is less or equal to d)

All integer partitions up to a given degree (that is, all integer partitions whose sum is less or equal to d), grouped by weight

Arguments

 :: (Int, Int) (height,width) -> [Partition]

All integer partitions fitting into a given rectangle.

Arguments

 :: (Int, Int) (height,width) -> [[Partition]]

All integer partitions fitting into a given rectangle, grouped by weight.

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

Arguments

 :: (Int, Int) (height,width) -> Int d -> [[Int]]

Integer partitions of d, fitting into a given rectangle, as lists.

Arguments

 :: (Int, Int) (height,width) -> Int d -> [Partition]

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

# Random partitions

randomPartition :: RandomGen g => Int -> g -> (Partition, g) Source

Uniformly random partition of the given weight.

NOTE: This algorithm is effective for small n-s (say n up to a few hundred / one thousand it should work nicely), and the first time it is executed may be slower (as it needs to build the table partitionCountList first)

Algorithm of Nijenhuis and Wilf (1975); see

• Knuth Vol 4A, pre-fascicle 3B, exercise 47;
• Nijenhuis and Wilf: Combinatorial Algorithms for Computers and Calculators, chapter 10

Arguments

 :: RandomGen g => Int number of partitions to generate -> Int the weight of the partitions -> g -> ([Partition], g)

Generates several uniformly random partitions of n at the same time. Should be a little bit faster then generating them individually.

# Dominance order

q `dominates` p returns True if q >= p in the dominance order of partitions (this is partial ordering on the set of partitions of n).

Lists all partitions of the same weight as lambda and also dominated by lambda (that is, all partial sums are less or equal):

dominatedPartitions lam == [ mu | mu <- partitions (weight lam), lam `dominates` mu ]

Lists all partitions of the sime weight as mu and also dominating mu (that is, all partial sums are greater or equal):

dominatingPartitions mu == [ lam | lam <- partitions (weight mu), lam `dominates` mu ]

# Partitions with given number of parts

Arguments

 :: Int k = number of parts -> Int n = the integer we partition -> [Partition]

Lists partitions of n into k parts.

sort (partitionsWithKParts k n) == sort [ p | p <- partitions n , numberOfParts p == k ]

Naive recursive algorithm.

Arguments

 :: Int k = number of parts -> Int n = the integer we partition -> Integer

# Partitions with only odd/distinct parts

Partitions of n with only odd parts

Partitions of n with distinct parts.

Note:

length (partitionsWithDistinctParts d) == length (partitionsWithOddParts d)

# Sub- and super-partitions of a given partition

Returns True of the first partition is a subpartition (that is, fit inside) of the second. This includes equality

This is provided for convenience/completeness only, as:

isSuperPartitionOf q p == isSubPartitionOf p q

Sub-partitions of a given partition with the given weight:

sort (subPartitions d q) == sort [ p | p <- partitions d, isSubPartitionOf p q ]

All sub-partitions of a given partition

Super-partitions of a given partition with the given weight:

sort (superPartitions d p) == sort [ q | q <- partitions d, isSubPartitionOf p q ]

# The Pieri rule

pieriRule :: Partition -> Int -> [Partition] Source

The Pieri rule computes s[lambda]*h[n] as a sum of s[mu]-s (each with coefficient 1).

See for example http://en.wikipedia.org/wiki/Pieri's_formula

The dual Pieri rule computes s[lambda]*e[n] as a sum of s[mu]-s (each with coefficient 1)

# ASCII Ferrers diagrams

Which orientation to draw the Ferrers diagrams. For example, the partition [5,4,1] corrsponds to:

In standard English notation:

@@@@@
@@@@
@

In English notation rotated by 90 degrees counter-clockwise:

@
@@
@@
@@
@@@

And in French notation:

@
@@@@
@@@@@

Constructors

 EnglishNotation English notation EnglishNotationCCW English notation rotated by 90 degrees counterclockwise FrenchNotation French notation (mirror of English notation to the x axis)

Instances

 Source Source

Synonym for asciiFerrersDiagram' EnglishNotation '@'

Try for example:

autoTabulate RowMajor (Right 8) (map asciiFerrersDiagram \$ partitions 9)