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partitions -- list the partitions of an integer

Synopsis

Description

i1 : partitions(4)

o1 = {Partition{4}, Partition{3, 1}, Partition{2, 2}, Partition{2, 1, 1},
     ------------------------------------------------------------------------
     Partition{1, 1, 1, 1}}

o1 : List
 
i2 : partitions(4, 2)

o2 = {Partition{2, 2}, Partition{2, 1, 1}, Partition{1, 1, 1, 1}}

o2 : List

Each partition is a basic list of type Partition.

i3 : p = new Partition from {2,2,1}

o3 = Partition{2, 2, 1}

o3 : Partition
 
i4 : member(p, partitions(5,2))

o4 = true
 
i5 : member(p, partitions(5,1))

o5 = false
 
i6 : conjugate(p)

o6 = Partition{3, 2}

o6 : Partition

For ordered lists of exactly k nonnegative integers that sum to n, use compositions instead. In the following example, we create Partition objects from the output of compositions to find all the partitions of 10, of length exactly 4, with no part greater than 5.

i7 : A = unique apply(compositions(4, 10), comp -> rsort comp);
 
i8 : B = select(A, a -> all(a, i -> 0<i and i<6));
 
i9 : apply(B, b -> new Partition from b)

o9 = {Partition{5, 3, 1, 1}, Partition{4, 4, 1, 1}, Partition{5, 2, 2, 1},
     ------------------------------------------------------------------------
     Partition{4, 3, 2, 1}, Partition{3, 3, 3, 1}, Partition{4, 2, 2, 2},
     ------------------------------------------------------------------------
     Partition{3, 3, 2, 2}}

o9 : List

If partitions n is called on a negative integer n, an empty list is returned. If a negative integer is given for k, it will cause an error.

See also

Ways to use partitions:

For the programmer

The object partitions is a method function.