Built-in Mathematica Symbol

IntegerPartitions

IntegerPartitions[n]
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, {k_min, k_max}]
gives partitions into between k_min and k_max integers.

IntegerPartitions[n, kspec, {s₁, s₂, ...}]
gives partitions involving only the s_i.

IntegerPartitions[n, kspec, sspec, m]
limits the result to the first m partitions.

MORE INFORMATION

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, {k_min, k_max, dk}] gives partitions into k_min, k_min+dk, ... integers.

n and the s_i can be rational numbers, and can be negative.

In the list of partitions, those involving earlier s_i are given last.

IntegerPartitions[n, kspec, sspec, -m] limits the result to the last m partitions.

In , a kspec of All corresponds to {0, Infinity}; an sspec of All corresponds to Range[n]; an m of All corresponds to Infinity.

All partitions of 5:

Scope (3)

Partitions of 8 into at most 3 integers:

Partitions of 8 into exactly 3 integers:

Find all partitions of 8 that involve only 1, 2 and 5:

Generalizations & Extensions (4)

Find partitions of 6 of even length only:

Find ways to form 3 from combinations of rational numbers:

Find partitions involving negative numbers:

Find the first 10 partitions of 15:

Find the last 3 partitions of 15:

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: