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.

Sorts the input, and cuts the nonpositive elements.

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

Assumes that the input is decreasing.

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

Converts a partition to an exponent vector.

For example,

toExponentVector (Partition [4,4,2,2,2,1]) == [1,3,0,2]

meaning (1^1,2^3,3^0,4^2).

This is simply the union of parts. For example

Partition [4,2,1] `unionOfPartitions` Partition [4,3,1] == Partition [4,4,3,2,1,1]

Note: This is the dual of pointwise sum, sumOfPartitions

Pointwise sum of the parts. For example:

Partition [3,2,1,1] `sumOfPartitions` Partition [4,3,1] == Partition [7,5,2,1]

Note: This is the dual of unionOfPartitions

Partitions of d.

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

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

All integer partitions fitting into a given rectangle.

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

Number of partitions of n (looking up a table built using Euler's algorithm)

Number of of d, fitting into a given rectangle. Naive recursive algorithm.

Count all partitions fitting into a rectangle. # = \binom { h+w } { h }

Count partitions of n into k parts.

Naive recursive algorithm.

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 of partitions counts 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

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

Dominance partial ordering as a partial ordering.

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 ]

Lists partitions of n into k parts.

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

Naive recursive algorithm.

Partitions of n with only odd parts

Partitions of n with distinct parts.

Note:

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

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 ]

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:

 @
 @@@@
 @@@@@

Synonym for asciiFerrersDiagram' EnglishNotation '@'

Try for example:

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