PartitionsQ

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

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguagePartitionsQ✓↗

PartitionsQ(n)$q(n)$, distinct-part partitions of $n$

SageMathPartitions(n, max_slope=-1).cardinality()✓↗

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.

Plot

Examples

Basic #

The partitions of 10 into distinct parts

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In$\mathrm{PartitionsQ}(10)$Out$10$

The empty partition

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In$\mathrm{PartitionsQ}(0)$Out$1$

Scope #

Exact for large n

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In$\mathrm{PartitionsQ}(100)$Out$444\,793$

Threads over a list

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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

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In$\mathrm{PartitionsQ}(\bigl\lbrack2, 4, 6\bigr\rbrack)$Out$\bigl\lbrack1, 2, 4\bigr\rbrack$

Properties #

Counts the DistinctPartitions family

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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

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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: $5,4+1,3+2$

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In$\mathrm{PartitionsQ}(5)$Out$3$

Only the partition $1$

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In$\mathrm{PartitionsQ}(1)$Out$1$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

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

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In$\mathrm{PartitionsQ}(2)$Out$1$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

$3$ and $2+1$

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In$\mathrm{PartitionsQ}(3)$Out$2$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

$4$ and $3+1$

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In$\mathrm{PartitionsQ}(4)$Out$2$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Four partitions into distinct parts

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In$\mathrm{PartitionsQ}(6)$Out$4$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Counts grow slower than unrestricted partitions

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In$\mathrm{PartitionsQ}(7)$Out$5$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Six partitions into distinct parts

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In$\mathrm{PartitionsQ}(8)$Out$6$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Eight partitions of 9 into distinct parts

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In$\mathrm{PartitionsQ}(9)$Out$8$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Equals the number of partitions of 11 into odd parts

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In$\mathrm{PartitionsQ}(11)$Out$12$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 15

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In$\mathrm{PartitionsQ}(12)$Out$15$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 18

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In$\mathrm{PartitionsQ}(13)$Out$18$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 22

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In$\mathrm{PartitionsQ}(14)$Out$22$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 27

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In$\mathrm{PartitionsQ}(15)$Out$27$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 32

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In$\mathrm{PartitionsQ}(16)$Out$32$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

The value 38

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In$\mathrm{PartitionsQ}(17)$Out$38$

Known value · Wolfram MathematicalFunctionData, PartitionsQ

Possible issues #

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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 agreeSageMath 21 agree1 errormathlib4 19 agree3 error
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

The partitions of n whose parts are all distinct (Nat.Partition.distincts), counted by enumerating them: exponential in n, so a large n runs into the per-item time cap.

See also: DistinctPartitions, IntegerPartitions