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--- author(s): Lily Silverstein
doc ///
Key
partitions
(partitions, ZZ)
(partitions, ZZ, ZZ)
Headline
list the partitions of an integer
Usage
partitions(n)
partitions(n, k)
Inputs
n:ZZ
a nonnegative integer
k:ZZ
a nonnegative integer, the maximum size of each part
Outputs
:List
of all partitions of {\tt n}
Description
Example
partitions(4)
partitions(4, 2)
Text
Each partition is a basic list of type @TO Partition@.
Example
p = new Partition from {2,2,1}
member(p, partitions(5,2))
member(p, partitions(5,1))
conjugate(p)
Text
For ordered lists of exactly {\tt k} nonnegative integers
that sum to {\tt n}, use @TO compositions@ instead. In the
following example, we create Partition objects from the
output of {\tt compositions} to find all the partitions of 10,
of length exactly 4, with no part greater than 5.
Example
--get list of unordered compositions without duplicates
A = unique apply(compositions(4, 10), comp -> rsort comp);
--select those with every part positive and no more than 5
B = select(A, a -> all(a, i -> 0<i and i<6));
--create Partition object from each one
apply(B, b -> new Partition from b)
Text
If {\tt partitions n} is called on a negative integer {\tt n},
an empty list is returned. If a negative integer is given
for {\tt k}, it will cause an error.
SeeAlso
compositions
(conjugate, Partition)
Partition
subsets
"combinatorial functions"
///
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