gives a list of all possible ways to partition the integer n into smaller integers.
IntegerPartitions [n,k]
gives partitions into at most k integers.
IntegerPartitions [n,{k}]
gives partitions into exactly k integers.
IntegerPartitions [n,{kmin,kmax}]
gives partitions into between kmin and kmax integers.
IntegerPartitions [n,kspec,{s1,s2,…}]
gives partitions involving only the si.
IntegerPartitions [n,kspec,sspec,m]
limits the result to the first m partitions.
IntegerPartitions
gives a list of all possible ways to partition the integer n into smaller integers.
IntegerPartitions [n,k]
gives partitions into at most k integers.
IntegerPartitions [n,{k}]
gives partitions into exactly k integers.
IntegerPartitions [n,{kmin,kmax}]
gives partitions into between kmin and kmax integers.
IntegerPartitions [n,kspec,{s1,s2,…}]
gives partitions involving only the si.
IntegerPartitions [n,kspec,sspec,m]
limits the result to the first m partitions.
Details
- Results from IntegerPartitions are normally given in reverse lexicographic order.
- Length [IntegerPartitions[n]] is PartitionsP [n].
- IntegerPartitions [n] is equivalent to IntegerPartitions [n,All ].
- IntegerPartitions [n,{kmin,kmax,dk}] gives partitions into kmin, kmin+dk, … integers.
- n and the si can be rational numbers, and can be negative.
- In the list of partitions, those involving earlier si are given last.
- IntegerPartitions [n,kspec,sspec,-m] limits the result to the last m partitions.
- In IntegerPartitions [n,kspec,sspec,m], a kspec of All corresponds to {0,Infinity }; an sspec of All corresponds to Range [n]; an m of All corresponds to Infinity .
Examples
open all close allBasic Examples (3)
All partitions of 5:
Partitions of 5 involving at most 3 terms:
Partitions of 5 involving at least 3 terms:
Scope (6)
All partitions of 8:
Partitions of 8 into at most 3 integers:
Equivalently:
Partitions of 8 into exactly 3 integers:
Find partitions of 8 of even length only:
Find all partitions of 8 that involve only 1, 2, and 5:
Find the first 10 partitions of 15:
Find the last 3 partitions of 15:
Generalizations & Extensions (2)
Find ways to form 3 from combinations of rational numbers:
Find partitions involving negative numbers:
Applications (2)
Find the ways to make change for 156 cents with 10 or fewer standard coins:
Find "McNugget partitions" for 50:
Find the number of "McNugget partitions" for numbers up to 50:
Show integers that are not "McNuggetable":
The last case is exactly the corresponding Frobenius number:
Properties & Relations (4)
Each sublist adds up to the original number:
The length of IntegerPartitions [n] is PartitionsP [n]:
IntegerPartitions gives results in reverse lexicographic order, not Sort order:
For integers below 10, generate IntegerPartitions order by converting to strings:
FrobeniusSolve gives coefficient lists for IntegerPartitions :
Possible Issues (3)
IntegerPartitions cannot give an infinite list of partitions:
There are no integer partitions of 1/2:
There are, however, partitions into rationals:
If all items requested by the fourth argument are not present, a warning message is issued:
To suppress the message, use Off :
See Also
PartitionsP Divisors Subsets IntegerDigits NumberDecompose FrobeniusSolve PowersRepresentations
Function Repository: IntegerPartitionQ IntegerCompositions NextIntegerPartition StrictIntegerCompositions DominatingIntegerPartitionQ IntegerPartitionFrequency FrobeniusSymbolFromPartition ConjugatePartition FerrersDiagram NumberOfTableaux StandardYoungTableaux TableauQ RandomIntegerPartition
Tech Notes
Related Guides
Related Links
Text
Wolfram Research (2007), IntegerPartitions, Wolfram Language function, https://reference.wolfram.com/language/ref/IntegerPartitions.html (updated 2008).
CMS
Wolfram Language. 2007. "IntegerPartitions." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/IntegerPartitions.html.
APA
Wolfram Language. (2007). IntegerPartitions. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/IntegerPartitions.html
BibTeX
@misc{reference.wolfram_2025_integerpartitions, author="Wolfram Research", title="{IntegerPartitions}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/IntegerPartitions.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_integerpartitions, organization={Wolfram Research}, title={IntegerPartitions}, year={2008}, url={https://reference.wolfram.com/language/ref/IntegerPartitions.html}, note=[Accessed: 16-November-2025]}