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--- author: Lily Silverstein
doc ///
Key
compositions
(compositions, ZZ)
(compositions, ZZ, ZZ)
Headline
list the compositions of an integer
Usage
compositions(k, n)
Inputs
k:ZZ
a nonnegative integer, the number of parts in each composition
n:ZZ
a nonnegative integer, the sum of each composition (if omitted, assumed to be also $k$)
Outputs
:List
of all ordered lists of {\tt k} nonnegative integers
that sum to {\tt n}
Description
Example
compositions(4, 2)
compositions(2, 4)
Text
To find all unordered compositions, we can use @TO sort@ or
@TO rsort@ to put each composition into a standard form, then
use the function @TO unique@ to remove duplicates.
Example
unique apply(compositions(4, 10), comp -> rsort comp)
Text
In the next example, we use @TO select@ to find all the compositions
of 18 into 5 parts, with each part greater than or equal to 3.
Example
select(compositions(5, 18), comp -> all(comp, c -> c>=3))
Text
For partitions of {\tt n} into unordered, {\em strictly positive} parts,
use @TO partitions@ instead.
If a negative integer is given for {\tt n},
an empty list is returned. If a negative integer is given
for {\tt k}, it will cause an error.
SeeAlso
partitions
subsets
///
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