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PermutationListQ[expr]

returns True if expr is a valid permutation list and False otherwise.

Details
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Examples  
Basic Examples  
Scope  
Properties & Relations  
Neat Examples  
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PermutationListQ

PermutationListQ[expr]

returns True if expr is a valid permutation list and False otherwise.

Details

  • A valid permutation list {p1,…,pn} is a rearrangement of the integers {1,…,n}.

Examples

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Basic Examples  (1)

A valid permutation list:

Wolfram Language code: PermutationListQ[{3, 2, 6, 1, 5, 4, 7, 8}]

Invalid permutation lists:

Wolfram Language code: PermutationListQ[{3, 2, 2, 4, 1}]
Wolfram Language code: PermutationListQ[{0, 1, 2, 3, 4}]
Wolfram Language code: PermutationListQ[{a, 1, 2, 3, 4}]

Scope  (2)

PermutationListQ works efficiently with large permutation lists:

Wolfram Language code: PermutationListQ[RandomSample[Range[1000000]]]

The empty list is considered a permutation list of length and degree 0:

Wolfram Language code: PermutationListQ[{}]

Properties & Relations  (4)

RandomSample[Range[n]] always gives a valid permutation list:

Wolfram Language code: RandomSample[Range[10]]
Wolfram Language code: PermutationListQ[%]

A possible, but less efficient, Wolfram Language implementation:

Wolfram Language code: permutationlistq[list_List] := SameQ[Union[list], Range[Length[list]]]
Wolfram Language code: permutationlistq[RandomSample[Range[10]]]

Validity of permutations in cyclic form is checked with PermutationCyclesQ. A permutation list can always be obtained as a permutation of the elements in canonical order using Permute:

Wolfram Language code: list = {1, 4, 3, 5, 7, 8, 10, 2, 6, 9}
Wolfram Language code: PermutationListQ[list]
Wolfram Language code: cycs = FindPermutation[Sort[list], list]
Wolfram Language code: PermutationCyclesQ[cycs]
Wolfram Language code: Permute[Sort[list], cycs] === list

Ordering always returns a permutation list, even if the elements of the expression are repeated:

Wolfram Language code: Ordering[head[a, a, b, c, b, a, c, b]]
Wolfram Language code: PermutationListQ[%]

Neat Examples  (1)

There are 409113 integer numbers up to whose decimal digits form permutation lists. This is how the first 153 (the largest being 54321) are distributed:

Wolfram Language code: ListPlot[Accumulate[PermutationListQ /@ IntegerDigits /@ Range[60000] /. {True -> 1, False -> 0}]]
Wolfram Research (2010), PermutationListQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationListQ.html.

Text

Wolfram Research (2010), PermutationListQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationListQ.html.

CMS

Wolfram Language. 2010. "PermutationListQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PermutationListQ.html.

APA

Wolfram Language. (2010). PermutationListQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PermutationListQ.html

BibTeX

@misc{reference.wolfram_2026_permutationlistq, author="Wolfram Research", title="{PermutationListQ}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationListQ.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_permutationlistq, organization={Wolfram Research}, title={PermutationListQ}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationListQ.html}, note=[Accessed: 13-June-2026]}