IntegerPartitions

IntegerPartitions(n)

returns all partitions of the integer n.

IntegerPartitions(n, k)

lists the possible ways to partition n into smaller integers, using up to k elements.

IntegerPartitions(n, {lower, upper}, {listOfIntegers})

lists the possible ways to partition n with the numbers in {listOfIntegers}, using between lower and upper number of elements.

See

• Wikipedia - Partition (number theory)
• Wikipedia - Coin problem
• Wikipedia - Diophantine equation

Examples

>> IntegerPartitions(3)
{{3},{2,1},{1,1,1}}

>> IntegerPartitions(10,2)
{{10},{9,1},{8,2},{7,3},{6,4},{5,5}}

The “McNugget partitions” OEIS - Number of partitions of n into parts 6, 9 or 20.

>> Table(Length(IntegerPartitions(i, All, {6, 9, 20})), {i,0, 100, 1})
{1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,1,1,0,0,2,0,1,2,0,1,2,0,1,2,0,1,3,0,2,2,
1,1,3,0,2,3,1,2,3,1,2,3,1,2,4,1,3,3,2,2,5,1,3,4,2,3,5,2,3,5,2,3,6,2,4,5,3,3,7,2,
5,6,3,4,7,3,5,7,3,5,8,3,6,7,4,5,9,3,7,8,5}

Related terms

FrobeniusNumber, FrobeniusSolve

Implementation status

• ✅ - full supported

Github

• Implementation of IntegerPartitions