PartitionsQ

The number $q(n)$ of partitions of $n$ into distinct parts.

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PartitionsQ(n)

q(n)

, distinct-part partitions of

n

Typed signatures

Type	Package	Applies
`(integer) -> integer`	`number-theory`	always

Domain: Number theory

Details

By Euler's theorem, $q(n)$ also counts the partitions of $n$ into odd parts — see IntegerPartitions.
Counts the DistinctPartitions family.

Examples

Basic #

The partitions of 10 into distinct parts

In

\mathrm{PartitionsQ}(10)

Out

10

The empty partition

In

\mathrm{PartitionsQ}(0)

Out

1

Scope #

Exact for large n

In

\mathrm{PartitionsQ}(100)

Out

444\,793

Threads over a list

In

\mathrm{PartitionsQ}(\bigl\lbrack1, 2, 3, 4, 5, 6, 7, 8\bigr\rbrack)

Out

\bigl\lbrack1, 1, 2, 2, 3, 4, 5, 6\bigr\rbrack

Threads over a list of arguments

In

\mathrm{PartitionsQ}(\bigl\lbrack2, 4, 6\bigr\rbrack)

Out

\bigl\lbrack1, 2, 4\bigr\rbrack

Properties #

Counts the DistinctPartitions family

In

\mathrm{PartitionsQ}(10)=\mathrm{Count}(\mathrm{DistinctPartitions}(10))

Out

\mathrm{True}

Euler: as many partitions into odd parts as into distinct parts, counted with IntegerPartitions' part restriction

In

\mathrm{PartitionsQ}(12)=\mathrm{Count}(\mathrm{IntegerPartitions}(12, \mathrm{All}, \bigl\lbrack1, 3, 5, 7, 9, 11\bigr\rbrack))

Out

\mathrm{True}

Partitions of 5 into distinct parts: $5,4+1,3+2$

In

\mathrm{PartitionsQ}(5)

Out

3

Only the partition $1$

In

\mathrm{PartitionsQ}(1)

Out

1

Known value · Wolfram MathematicalFunctionData, PartitionsQ

$2$ alone, since $1+1$ repeats a part

In

\mathrm{PartitionsQ}(2)

Out

1

Known value · Wolfram MathematicalFunctionData, PartitionsQ

$3$ and $2+1$

In

\mathrm{PartitionsQ}(3)

Out

2

Known value · Wolfram MathematicalFunctionData, PartitionsQ

$4$ and $3+1$

In

\mathrm{PartitionsQ}(4)

Out

2

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Four partitions into distinct parts

In

\mathrm{PartitionsQ}(6)

Out

4

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Counts grow slower than unrestricted partitions

In

\mathrm{PartitionsQ}(7)

Out

5

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Six partitions into distinct parts

In

\mathrm{PartitionsQ}(8)

Out

6

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Eight partitions of 9 into distinct parts

In

\mathrm{PartitionsQ}(9)

Out

8

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Equals the number of partitions of 11 into odd parts

In

\mathrm{PartitionsQ}(11)

Out

12

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 15

In

\mathrm{PartitionsQ}(12)

Out

15

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 18

In

\mathrm{PartitionsQ}(13)

Out

18

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 22

In

\mathrm{PartitionsQ}(14)

Out

22

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 27

In

\mathrm{PartitionsQ}(15)

Out

27

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 32

In

\mathrm{PartitionsQ}(16)

Out

32

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 38

In

\mathrm{PartitionsQ}(17)

Out

38

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Possible issues #

In

\mathrm{PartitionsQ}(-\mathrm{Expand}((e+1)^2)+12+(1+e)^2)

Out

\mathrm{PartitionsQ}(11-2e-e^{2}+(1+e)^2)

About

Examples	24
Checked against	Wolfram Language 23 agree
Crosswalk	Wolfram Language

See also: DistinctPartitions, IntegerPartitions